Methods and apparatus for error correction

ABSTRACT

An error correction decoder which includes a syndrome calculator, an error locator polynomial calculator, a standard error locator polynomial calculator, an error transform calculator and an inverse error transform calculator. These error correction decoder calculators provide a pipelined architecture for performing Reed-Solomon error correction calculations quickly. The error locator polynomial calculator includes an R-Q calculator, a λ-μ calculator, an R-Q degree calculator and a trigger circuit. These calculators and the trigger circuit can be implemented each as a plurality of generic cells. The number of generic cells can be changed to construct Reed-Solomon circuits for different Reed-Solomon codes. The R-Q, λ-μ and R-Q degree calculators provide adaptive circuits that use switches and multiplexors, for example, to adapt to perform appropriate calculations based upon the nature of the error correction polynomials applied to the inputs of the calculators. The R-Q, λ-μ and R-Q degree calculators use multipliers, adders, memory elements and/or delay elements to perform the appropriate calculations. The calculations performed are controlled by selecting a path through which data will pass wherein the path is configured to perform the appropriate calculations. The R-Q degree calculator is similar. The trigger circuit provides an adaptive delay using a multiplexor, a bypass path and a delay path. The trigger circuit adjusts the timing of the trigger signal it puts out by selecting either the bypass path or the delay path. By adjusting the delay, the trigger circuit coordinates the triggering of subsequent cells with the timing of calculations performed in preceding cells.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The field of the invention relates generally to error correction andmore particularly to a pipeline decoder that can be used to decodeReed-Solomon codes, for example.

2. Background

Today's digital systems, such as data transmission and storage systems,typically need to be efficient and reliable. As large, high speed datanetworks become more widely used and integral with day to dayactivities, individuals likely will depend more upon the efficient andreliable reproduction of the data that these networks transmit. Toachieve such an efficient and reliable operation, digital systemstypically use techniques for controlling errors and to ensure reliabledata reproduction. Reed Solomon ("RS") codes have been used in digitalsystems to provide error control, including both error detection anderror correction. Such systems have included, for example, spacetelecommunication systems, compact disc systems and digitaltelecommunication systems.

A Reed Solomon code consists of a set of code words. Each code wordcontains data and error correcting information. A code word ischaracterized by a total number of symbols N which include data symbolsand parity symbols. Each symbol is m bits of binary data where m mightbe 8, for example. In a Reed Solomon code that uses m bit symbols, N isoften chosen such that N=2^(m) -1. Of the total number of symbols N, Kof the symbols are data symbols. Accordingly, N-K of the symbols areparity symbols. Reed Solomon codes are typically designed to detect ordetect and correct a predetermined number of errors. A code designed todetect and correct up to t errors in each N symbol code word, forexample, typically requires K=2t parity symbols (bits). Reed Solomoncodes are commonly described as (N,K) Reed Solomon codes to communicatethe values N and K of the particular code.

Reed Solomon codes work by encoding data according to certainpredetermined rules to produce encoded data having a known format. Theencoded data might then be transmitted to a receiver, for example. Thedata that is transmitted shall be referred to as the transmitted data.The receiver, knowing the predetermined rules and the known format,often can detect errors in the received data by comparing it to theknown format. If any differences exist, an error has occurred. Usingcertain rules, the receiver often can correct errors detected in thismanner. After such correction, the corrected data can then be convertedback from its encoded format to obtain the original data. The process ofdetecting or detecting and correcting errors and converting the encodeddata back to its original form is called decoding. See Lin, S. andCostello, D. Jr., Error Control Coding: Fundamentals and Applications,Prentice Hall (1983) for a discussion of Reed Solomon error correctionand its related principles.

Numerous calculations can be required to perform the encoding anddecoding processes associated with RS codes. In addition, thesecalculation often must be performed in real time. Accordingly, RSencoders and decoders often must meet certain minimum performancelevels. With respect to RS decoders, efforts have been made to producealgorithms and/or techniques that enable the calculations to beperformed quickly. Unfortunately, increasing the speed of RS decoderscan also increase the cost.

For example, the speed of a decoder might be enhanced by using parallelarchitectures or large memory arrays. A parallel architecture mayinvolve implementing the same functions in multiple areas of a chip.These multiple areas may be used, for example, when the same function,such as a calculation, needs to be performed repeatedly. Often, therepeated functions can be accomplished simultaneously in the multipleareas saving time. Unfortunately, implementing the same function inmultiple areas typically requires more chip area than simply using asingle chip area repeatedly to perform the single function. This latterapproach, while possibly saving chip area, can result in delay becausethe one chip area typically will have to perform the function multipletimes. A batch of data that is to be processed by the chip area may haveto wait until the chip area is done performing the function on aprevious batch of data. Additionally, such repeated use might requireadditional control circuitry to handle timing or feedback issues, forexample, and as a result additional chip space. Such a design might alsorequire additional clock cycles and possibly additional buffers to storeinterim calculations.

Memory arrays might be used by an RS decoder to store data or look uptables that might eliminate some of the needed calculations, forexample. Memory arrays, however, can increase the amount of chip areaused by a decoder, resulting in a more expensive decoder. The chip arearequired by an RS decoder might be reduced by avoiding such parallelarchitectures or memory arrays. Such a decoder might be less expensive,but it may not provide the desired speed.

Accordingly, there has been a need for a Reed-Solomon decoder thatobtains the desired processing speed without unacceptably increasing thechip area used by or the cost of the decoder.

SUMMARY OF THE INVENTION

An aspect of the present invention provides a λ-μ calculator that has apipelined structure for performing Reed Solomon error correctioncalculations quickly. Another aspect of the invention provides a λ-μcalculator cell that affords flexibility in designing a λ-μ calculator.In particular, an aspect of the invention provides a λ-μ calculator cellthat can, but need not, be used in duplicate to construct λ-μcalculators for a wide variety of Reed Solomon codes that have varyingnumbers of data and parity bytes. An aspect of the present inventionprovides a λ-μ calculator cell that can be used to minimize the need formemory arrays and look-up tables. An aspect of the present inventionprovides a λ-μ calculator that efficiently uses the area of asemiconductor die. An aspect of the present invention provides a λ-μcalculator cell that can be used to minimize wait time associated withusing a single area of a chip to perform repeated error correctioncalculations.

In an embodiment of the invention, a λ-μ calculator is provided forprocessing error correction polynomials in an error correction decoderdesigned to detect t errors. This embodiment includes error correctiondecoders that detect and correct t errors. The λ-μ calculator includes aplurality of λ-μ circuits each having a λ input, a λ output, a μ inputand a μ output. The plurality of λ-μ circuits are coupled together inseries such that the λ output of each immediately preceding λ-μ circuitis coupled to the λ input of each immediately succeeding λ-μ circuit andthe μ output of each immediately preceding λ-μ circuit is coupled to theμ input of each immediately succeeding λ-μ circuit. The λ input of afirst of the λ-μ circuits provides an initial λ input, the μ input ofthe first λ-μ circuit provides an initial μ input, the λ output from alast of the λ-μ circuits provides a final λ output and the μ output fromthe last λ-μ circuit provides a final μ output.

In an embodiment of the invention, each λ-μ circuit includes a switchhaving a λ switch input, a μ switch input, a λ' switch output, a μ'switch output and a switch control input. The switch is adapted tocouple the λ switch input to the λ' switch output and the μ switch inputto the μ' switch output when a first control signal is present at theswitch control input. The switch is adapted to couple the λ switch inputto the μ' switch output and the μ switch input to the λ' switch outputwhen a second control signal is present at the switch control input. Ineach λ-μ circuit, the λ switch input is coupled to the λ input and the μswitch input is coupled to the μ input. In each λ-μ circuit a λprocessing path is coupled in series between the λ' switch output and aλ bypass multiplexor wherein the λ processing path is adapted to processerror correction polynomials. In each λ-μ circuit a λ processing bypasspath is coupled in series between the λ input and the λ bypassmultiplexor wherein the λ processing bypass path is adapted to passerror correction polynomials unprocessed. The λ bypass multiplexor isadapted to couple alternatively the λ input to the λ output through theλ processing bypass path and the λ' switch output to the λ outputthrough the λ processing path.

In each λ-μ circuit, a μ processing path is coupled in series betweenthe μ' switch output and a μ bypass multiplexor wherein the μ processingpath is adapted to process error correction polynomials. In each λ-μcircuit a μ processing bypass path is coupled in series between the μinput and the μ bypass multiplexor wherein the μ processing bypass pathis adapted to pass error correction polynomials unprocessed. The μbypass multiplexor is adapted to couple alternatively the μ input to theμ output through the μ processing bypass path and the μ' switch outputto the μ output through the μ processing path.

In an embodiment of the invention, the λ processing path includes a λ'delay path, a λ' calculation path, a λ' multiplexor and a delayadjustment circuit. The λ' delay path is coupled in series with the λ'multiplexor, the delay adjustment circuit and the λ bypass multiplexorbetween the λ' switch output and the λ output. The λ' calculation pathis coupled in series with the λ' multiplexor, the delay adjustmentcircuit and the λ bypass multiplexor between the λ' switch output andthe λ output. The λ' multiplexor is adapted to couple the λ' switchoutput to the λ output through one of the λ' delay path and the λ'calculation path.

In an embodiment of the invention, the delay adjustment circuit in eachλ-μ circuit includes a delay adjustment multiplexor, a delay elementcoupled to the delay adjustment multiplexor, and a delay bypass pathcoupled to the delay adjustment multiplexor. The delay adjustmentmultiplexor is adapted to couple the λ' switch output to the λ outputthrough one of the delay element and the delay bypass path. The delayelement can be a memory element. In this embodiment, the delay elementprovides a delay that is longer than a delay provided by the delaybypass path. In an embodiment, the delay provided by the delay elementis at least one cycle longer than a delay provided by the delay bypasspath. In an embodiment of the invention, an R polynomial is associatedwith a particular λ-μ circuit. When a degree of the R polynomialassociated with the particular λ-μ circuit is less than t, then thedelay adjustment multiplexor couples the λ' switch output to the λoutput through the delay element. When the degree of the R polynomialassociated with the particular λ-μ circuit is not less than t, then thedelay adjustment multiplexor couples the λ' switch output to the λoutput through the delay bypass path.

In an embodiment of the invention, the μ processing path in each λ-μcircuit includes a μ' delay path, a μ' bypass path and a μ' multiplexor.In this embodiment, the μ' delay path is coupled in series with the μ'multiplexor between the μ' switch output and the μ output and the μ'bypass path is coupled in series with the μ' multiplexor between the μ'switch output and the μ output such that the μ' multiplexor is adaptedto couple the μ' switch output to the μ output through one of the μ'delay path and the μ' bypass path. In an embodiment of the invention, across couple path is coupled between the μ' delay path and the λ'calculation path in each λ-μ circuit.

In an embodiment of the invention, the λ bypass multiplexor couples theλ input to the λ output through the λ processing bypass path when the μbypass multiplexor couples the μ input to the μ output through the μprocessing bypass path. In an embodiment of the invention, the λ bypassmultiplexor couples the λ' switch output to the λ output through the λprocessing path when the μ bypass multiplexor couples the μ' switchoutput to the μ output through the μ processing path.

In an embodiment of the invention, the λ' multiplexor couples the λ'switch output to the λ output through the λ' calculation path when theλ' multiplexor couples the μ' switch output to the μ output through theμ' delay path. In this embodiment, the λ' delay path introduces a delaythat is greater than a delay introduced by the λ' calculation path. Thedelay introduced by the μ' delay path may be at least one cycle longerthan the delay introduced by the λ' calculation path. The μ' delay pathmay be implemented using a memory element.

In an embodiment of the invention, the λ' multiplexor couples the λ'switch output to the λ output through the λ' delay path when the μ'multiplexor couples the μ' switch output to the μ output through the μ'bypass path. In this embodiment, the λ' delay path introduces a delaythat is longer than a delay introduced by the μ' bypass path. The delayintroduced by the λ' delay path may be at least one cycle longer thanthe delay introduced by the μ' bypass path. The λ' delay path may beimplemented using a memory element.

In an embodiment of the invention, a delay element and/or a memoryelement may, but need not, be coupled in series between the λ input andthe λ switch input such that the delay element and/or the memory elementare also coupled in series with the λ processing bypass path between theλ input and the λ bypass multiplexor. In an embodiment of the invention,a second delay element and/or a second memory element may, but need not,be coupled in series between the μ input and the μ switch input suchthat the second delay element and/or the second memory element are alsocoupled in series with the μ processing bypass path between the μ inputand the μ bypass multiplexor.

In an embodiment of the invention, the μ' bypass path, the μ'multiplexor and the μ bypass multiplexor are adapted to couple directlythe μ' switch output to the μ output.

In an embodiment of the invention, an R polynomial and a Q polynomialare associated with a particular λ-μ circuit. When a degree of one ofthe R polynomial and the Q polynomial that are associated with theparticular λ-μ circuit is less than t, then the λ bypass multiplexor inthe particular λ-μ circuit couples the λ input to the λ output throughthe λ processing bypass path and the μ bypass multiplexor in theparticular λ-μ circuit couples the μ input to the μ output through the μprocessing bypass path. When the degree of the R polynomial that isassociated with the particular λ-μ circuit is not less than t and whenthe degree of the Q polynomial that is associated with the particularλ-μ circuit is not less than t, then the λ bypass multiplexor in theparticular λ-μ circuit couples the λ input to the λ output through the λprocessing path and the μ bypass multiplexor in the particular λ-μcircuit couples the μ input to the μ output through the μ processingpath.

In an embodiment of the invention, a first Q polynomial and a second Qpolynomial are associated with a particular λ-μ circuit. When a leadcoefficient of the first Q polynomial that is associated with theparticular λ-μ circuit is zero and when a degree of the second Qpolynomial that is associated with the particular λ-μ circuit is notless than t, then the λ' multiplexor couples the λ' switch output to theλ output through the λ' delay path and the μ' multiplexor couples the μ'switch output to the μ output through the μ' bypass path. When the leadcoefficient of the first Q polynomial that is associated with theparticular λ-μ circuit is not zero or when the degree of the second Qpolynomial that is associated with the particular λ-μ circuit is lessthan t, the λ' multiplexor couples the λ' switch output to the λ outputthrough the λ' calculation path and the μ' multiplexor couples the μ'switch output to the μ output through the μ' delay path.

In an embodiment of the invention, an R polynomial and a Q polynomialare associated with a particular λ-μ circuit. When a degree of the Rpolynomial that is associated with the particular λ-μ circuit is notless than a degree of the Q polynomial that is associated with theparticular λ-μ circuit, then the switch in the particular λ-μ circuitcouples the λ switch input to the λ' switch output and the μ switchinput to the μ' switch output. When the degree of the R polynomial thatis associated with the particular λ-μ circuit is less than the degree ofthe Q polynomial that is associated with the particular λ-μ circuit,then the switch in the particular λ-μ circuit couples the λ switch inputto the μ' switch output and the μ switch input to the λ' switch output.

In an embodiment of the invention, a cross couple path is coupledbetween the μ processing path and the λ' calculation path. The λ'calculation path includes a multiplier having a first input, a secondinput and an output, a calculation path memory element having an outputand an adder having a first input, a second input and an output. The λ'switch output is coupled to the first input of the multiplier, theoutput of the calculation path memory element is coupled to the secondinput of the multiplier, the output of the multiplier is coupled to thefirst input of the adder, the cross couple path is coupled to the secondinput of the adder and the output of the adder is coupled to the λ'multiplexor.

In an embodiment of the invention, the cross couple path includes across couple memory element having an output and a multiplier having afirst input, a second input and an output. The output of the crosscouple memory element is coupled to the first input of the multiplier,the μ processing path is coupled to the second input of the multiplierand the output of the multiplier is coupled to the second input of theadder.

In an embodiment of the invention, a polynomial output from the λ'switch output of the particular λ-μ circuit is the λ' polynomial forthat particular λ-μ circuit and a polynomial output from the μ' switchoutput of the particular λ-μ circuit is the μ' polynomial for thatparticular λ-μ circuit. The calculation path memory element in theparticular λ-μ circuit stores the leading coefficient of the Q'polynomial and the cross couple memory element in the particular λ-μcircuit stores the leading coefficient of the R' polynomial from anassociated R-Q circuit.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a generalized system 100 in which anembodiment of the invention is used to detect and correct errors;

FIG. 2 is a block diagram of a more specific system 200 in which anembodiment of the present invention is used to detect and correcterrors;

FIG. 3 is a block diagram of the decoder 202 used in system 200;

FIG. 4 illustrates a syndrome calculator 304 used by the decoder of FIG.3;

FIG. 5 illustrates a functional block diagram of an error locatorpolynomial calculator 310 used by the decoder of FIG. 3;

FIG. 6 illustrates an R-Q calculator cell 514 used in the error locatorpolynomial calculator of FIG. 5;

FIG. 7 illustrates a trigger circuit 516 used in the error locatorpolynomial calculator of FIG. 5;

FIG. 8 illustrates an R-Q degree calculator cell 512 used in the errorlocator polynomial calculator of FIG. 5;

FIG. 9 illustrates the λ-μ calculator cell 518 used in the error locatorpolynomial calculator of FIG. 5;

FIG. 10 illustrates the standard error-locator polynomial calculatorcircuit 312 used by the decoder of FIG. 3;

FIG. 11A illustrates the error transform calculator 314 used by thedecoder of FIG. 3;

FIG. 11B illustrates the trigger circuit associated with the errortransform calculator 314 of FIG. 11A; and

FIG. 12 illustrates the inverse error transform calculator used by thedecoder of FIG. 3;

FIG. 13 is an embodiment of a Galois field multiplier that can be usedin embodiments of the present invention;

FIGS. 14A-14F illustrate functional diagrams of a normal to dual basisconverter that may be used in the Galois field multiplier of FIG. 13;

FIGS. 15A-15B illustrate a mapping between a normal basis and a dualbasis of the Galois field GF(2⁸);

FIG. 16A the derivation of dual basis vectors used by an embodiment ofthe invention;

FIG. 16B illustrates additional dual basis vectors generated asdescribed an illustrated with reference to FIG. 16A;

FIG. 16C illustrates vector multiplications used to generate bits of thedual basis product GZ_(d) ;

FIGS. 17A-17G illustrate functional diagrams of a dual to normal basisconverter that may be used in the Galois field multiplier of FIG. 13;

FIG. 18 illustrates an example of a circuit used to implement a vectormultiplier circuit of an embodiment of the present invention.

DETAILED DESCRIPTION

Embodiments of the invention provide a novel Reed Solomon decoder. Thefollowing description is presented to enable a person skilled in the artto make and use the invention. Descriptions of specific embodiments areprovided only as examples. Various modifications to the describedembodiments may be apparent to those skilled in the art, and the genericprinciples defined herein may be applied to other embodiments andapplications without departing from the spirit and scope of theinvention. Thus, the present invention is not intended to be limited tothe described or illustrated embodiments, but is to be accorded thewidest scope consistent with the principles and features disclosedherein.

FIG. 1 illustrates a generalized block diagram of a typical system 100that uses an embodiment of the present invention. In system 100, digitalinformation 102 originates from a digital information source 104. Inthis description, information from an information source shall bereferred to generally as I(x). The source 104 might be a file stored ona network server, an application program, an A/D converter processing ananalog signal or any other source generating digital information. Thedigital information 102 is fed into the error correction encoder 106which puts the digital information into an encoded format. The encodeddigital information 108 is then transmitted or stored, for example, asrepresented schematically by channel 110. In this description, thetransmitted or stored information that comes out of an error correctionencoder shall be referred to generally as t(x) where t(x)=t₀ x^(N-1) +t₁x^(N-2) +. . . +t_(N-1) x⁰. Channel 110 schematically represents any ofa number of functions or operations that might be performed on the dataafter it comes out of the encoder 106 and before it is passed into thedecoder 112. Channel 110 might represent, for example, modulating theinformation t(x), passing it over a transmission medium and demodulatingit. In the alternative or in addition, it might represent encapsulatingand de-encapsulating the information t(x) using a variety of networkingprotocols, for example. It might include writing to a disk and accessfrom a disk, for example. Noise 116 represents any noise that mightaffect the information 108 in channel 110.

After passing through channel 110, the encoded digital information 108'passes to the error correction decoder 112. The digital information 108'will be the same as digital information 108 unless noise 116 has causedan error. In this description, the information received from a channelsuch as channel 110 at the input of an error correction decoder shall bereferred to generally as the received information r(x) where r(x)=r₀x^(N-1) +r₁ x^(N-2) +. . . +r_(N-1) x⁰. Error correction decoder 112,which is an embodiment of the present invention, detects and correctserrors, if possible, recreating the information stream t(x). Decoder 112then converts this encoded information stream t(x) back to the digitalinformation 102. Digital information 102 can then be provided to theinformation destination 114, which might be a client on a network, forexample, or any other destination to which the information I(x) is beingcommunicated.

FIG. 2 is a conceptual block diagram of a more specific system 200 thatuses a decoder 202. Decoder 202 is an embodiment of the presentinvention. System 200 is generally disclosed in EuropeanTelecommunication Standard (ETS) 300 429. Because the system 200 isintended to error correct a 204 byte MPEG packet (which includes paritybytes), it uses a (204, 188) Reed Solomon code. In particular, system200 is designed to correct up to 8(t) erroneous symbols of informationin every 204 symbols. Accordingly, 16(2t) parity symbols are used. Eachsymbol is a byte of information or an 8 bit word (m). The (204, 188)Reed Solomon code used by the system 200 is a shortened version of the(255,239) Reed Solomon Code and has the code generator polynomial

g(x)=(x+α⁰) (x+α¹) (x+α²) (x+α³) . . . (x+α¹⁵) where α is 02hex

and the field generator polynomial

p(X)=x⁸ +x⁴ +x³ +x² +1.

Because these polynomials are associated with a (255, 239) Reed Solomoncode, the Reed Solomon hardware used could be easily modified to handleup to 239 data bytes. Because the system 200 only uses 188 data bytes,however, the system 200 simply does not use the additional 51 data words(i.e. 239-188 unused data words). Other embodiments of the invention canuse other Reed Solomon codes or even other non-Reed Solomon errorcorrecting codes. To do so, the present embodiment will have to beadapted to provide the appropriate calculations for the particular codebeing used.

In the system 200, the 188 bytes of pre-encoded information areconverted to 204 bytes of coded information by encoder 204 to producethe transmitted data t(x), which is a (204, 188) RS code word. In thepresent embodiment, however, N=204 and K=16 because this code is ashortened version of the (255,239) Reed Solomon code. As illustrated inFIG. 2, the transmitted data t(x) is passed through blocks 206, overchannel 208 and through blocks 210 to the input of Reed-Solomon decoder202. Blocks 206, channel 208 and blocks 210 are a specific example ofthe channel 110 described in FIG. 1.

Decoder 202, which is an embodiment of the present invention, is a (204,188) RS decoder. The data r(x) received at the input of decoder 202 is a(204, 188) RS code word that may or may not contain errors. Thisinformation is decoded by decoder 202 to reproduce the information I(x)that was input to the RS encoder 204. Decoder 202 could easily beadapted to operate on (255, 239) Reed Solomon code words.

FIG. 3 is an expanded view of the decoder 202 that is used in the system200. Again t(x) represents the information transmitted by encoder 204and r(x) represents the information received at the input of decoder202. Elements 206, 208 and 210 of FIG. 2 are represented schematicallyin FIG. 3. The decoder 202 receives the received information r(x)through input 302. In the present embodiment r(x)=r₂₀₃ x²⁰³ +r₂₀₂ x²⁰²+r₂₀₁ x²⁰¹ +. . . +r₀. This received information r(x) passes to themodulo two adder 306 through delay 308 and passes to the syndromecalculator 304. The delay 308 is chosen so that the coefficients of thereceived polynomial r(x) are provided to the modulo two adder 306 to bemodulo two added to the corresponding coefficients of the inverse errortransform polynomial e(x). Accordingly, the delay 308 provides a delayequal to the time it takes the error locator polynomial calculator 310,the standard error locator polynomial calculator 312, the errortransform calculator 314 and the inverse error transform calculator 316to calculate the coefficients of the inverse error transform polynomiale(x) that correspond to the coefficients of r(x). In particular, delay308 is designed such that rn and en are applied to the inputs of themodulo two adder 306 to produce the output rn+en. This output is clockedinto the converter 318 where it is converted back to I(x). When dealingwith systemic codes such as Reed Solomon codes, the process ofconverting the error corrected data back to the information I(x) issimply a matter of dropping parity bytes which are in known locations.Accordingly, converter 318 in the present embodiment is simply a devicethat drops the parity bytes.

The decoder 202 detects and corrects errors in the received data streamr(x) in the following manner. In particular, syndrome calculator 304calculates the syndrome polynomial S(x) from the received polynomialr(x) using the relation ##EQU1## where S_(k) is the coefficient of thedegree k term of the polynomial S(x), r_(n) is the coefficient of thedegree n term of the received polynomial r(x) and α is a root of thefield generator polynomial p(x). Again, in the present embodimentα=02_(hex). In the present embodiment, ##EQU2## 15 due to the shortenedRS code. Accordingly, S(x)=S₀ x¹⁵ +S₁ x¹⁴ +S₂ x¹³ +. . . +S.sub.₁₅. Thesyndrome polynomial S(x) is passed to the error locator polynomialcalculator 310 and is passed through a delay 320 to the error transformcalculator 314. The delay 320 corresponds to the time it takes the errorlocator polynomial calculator 310 and the standard error locatorpolynomial calculator 312 to calculate the coefficients of the standarderror locator polynomial σ(x). In particular, the coefficients of thesyndrome polynomial S(x) and the coefficients of the standard errorlocator polynomial σ(x) are timed to arrive at the error transformcalculator 314 as appropriate to calculate the error transformpolynomial E(x).

The error locator polynomial calculator uses the iterative modifiedEuclid's algorithm to calculate the error locator polynomial λ(x) fromthe syndrome polynomial S(x) The polynomial λ(x) can have a maximumdegree of t, which in the present embodiment is 8. Accordingly,

    λ(x)=λ.sub.0 x.sup.8 +λ.sub.1 x.sup.7 +λ.sub.2 x.sup.6 + . . . λ.sub.8.

The standard error locator polynomial calculator 312 calculates σ(x)from λ(x). Thus,

    σ(x)=σ.sub.0 x.sup.8 +σ.sub.1 x.sup.7 +σ.sub.2 x.sup.6 + . . . σ.sub.8

where σ_(k) =λ_(k) /λ_(lnz) 0≦k≦8 where σ_(k) represents thecoefficients of σ(x), λ_(k) represents the same degree coefficient ofλ(x), λ_(lnz) is the leading (i.e., highest degree) non-zero coefficientof λ(x) and "lnz" (the subscript of λ_(lnz)) is the position of thisleading non-zero coefficient. Thus, if λ₂ is the leading non-zerocoefficient of the error locator polynomial, then Inz=2.

The error transform calculator 314 calculates the error transformpolynomial E(x) from the syndrome polynomial S(x) and from the standarderror locator polynomial σ(x) using the relation ##EQU3## where σ_(k) isthe coefficient of the degree 8-k (i.e. t-k) term of the standard errorlocator polynomial σ(x) and E_(2t+j-k+lnz) is a known coefficient of thepolynomial E(x). In the present embodiment, ##EQU4## The syndromes S₀through S₁₅ are the error transforms E₀ through E₁₅, respectively.Accordingly, the error transform calculator starts with these knownerror transforms, E₀ through E₁₅, and calculates the remaining errortransforms E₁₆ to E₂₅₄.

The inverse error transform calculator 316 then calculates the inverseerror transform polynomial e(x) from the error transform polynomial E(x)using the relation ##EQU5## where E_(k) is the coefficient of the degreek term of the error transform polynomial E(x). In the presentembodiment, ##EQU6## even though N-1=254 because of the use of theshortened Reed Solomon code. Inverse error transforms e₂₀₄ to e₂₅₄ neednot be calculated because they relate to errors in the 51 unused databytes. Other embodiments might calculate these inverse error transforms.The output of the inverse error transform calculator 316 is coupled toan input of the modulo two adder 306.

The modulo two adder 306 adds the coefficients of the received datastream r(x) to the corresponding coefficients of the inverse errortransform polynomial e(x). Assuming there are t errors or less in r(x),this addition typically corrects errors in r(x) to reproduce theinformation stream t(x) at the output of the modulo two adder 306. Thisinformation stream t(x) is then converted by converter 318 from the(204, 188) RS encoded code words back into the actual information streamI(x) that was present at the input of the Reed Solomon encoder 204 byremoving the parity bytes from the data bytes.

All of the bus lines illustrated in FIG. 3 are 8 bit wide bus lines. Inthis specification, "bus" or "bus line" shall generally refer to anypath for coupling a signal, such as a data signal or a control signal,between hardware elements such as the hardware elements of the decoder202. The hardware elements of decoder 202 include the componentsillustrated in FIG. 3 and the elements that make up these componentssuch as memory elements, multiplexors, adders, multipliers, switches,delays, cells or control circuits, for example. Bus lines include, forexample, control lines, inputs, outputs and lines coupling hardwareelements to each other.

Syndrome Calculator

FIG. 4 illustrates the syndrome calculator 304 used by decoder 202.Other embodiments of the invention can use other syndrome calculators.This syndrome calculator uses 16(2t) syndrome cells 402A to 402P tocalculate the equation ##EQU7## Each cell calculates one coefficientS_(k) of S(x). Embodiments of the invention that deal with syndromepolynomials having more or less terms can vary the number of cells tocorrespond to different number of terms. Other embodiments might use asingle cell repeatedly or various combinations of parallel cells andrepeated cells. Cells 402B to 402N have been illustrated symbolicallyfor convenience. In this specification, features of the presentembodiment that are described with an appended letter, such as cells402A to 402P, shall be collectively and generically referred to by thenumber without the appended letter. For example, "cells 402" shall referto all of the cells 402A to 402P collectively. "Cell 402" shallgenerically refer to any of the cells 402A to 402P. The term cell, asused herein, is not intended to require that the elements "within" acell be located in a common physical area of a chip, for example.Rather, the term cell is intended to describe a group of hardwareelements that have functional and/or physical relationships between themthat are similar to the functional and/or physical relationships betweenthe hardware elements in another group (i.e., in another cell).Calculator 304 includes a received data input 448, a multiplexor 436,logic zero inputs 444, 446 and 456, a multiplexor 438 and an output 446.All of the bus lines in the calculator 304 except for control lines 442and 434 are 8 bits wide. Controls lines 442 and 434 are one bit lines.

Each of the cells 402 includes an input memory element 408 (i.e. 408A to408P). In the present embodiment, the memory elements 408 are 8 bitregisters that provide the data that they are storing to their outputbuses 410. When each memory element 408 is clocked by a clock signal(not shown), it stores the data that is present at its input. The clocksignals that clock the memory elements in this specification may beprovided in any manner consistent with the operation of the decoder 202described herein.

Each of the cells 402 includes a power of a memory element 412 (i.e.412A to 412P) and a sum memory element 416 (i.e. 416A to 416P). Similarto memory elements 408, memory elements 412 and 416 are 8 bit registersthat provide the data that they are storing to their outputs. The memoryelement 412 has one output 414. The memory element 416 has two outputs418 and 420. When each element 416 is clocked by a clock signal (notshown), it stores the data that is present at its input. Memory elements412 store increasing powers of α, the powers ranging from 0 to 15(2t-1). In particular, α⁰ is stored in memory element 412A, α¹ is storedin memory element 412B and α² is stored in memory element 412C. α³ toα¹⁵ are stored in memory elements 412D to 412P, respectively. In thepresent embodiment, the memory elements have the powers of α burned intotheir registers to provide a fast, parallel design. In alternateembodiments, these memory elements could clock in the powers of α infrom an input or they could use some form of ROM.

Each of the cells 402 includes a multiplier 422 (i.e. 422A to 422P).These multipliers 422 are Galois field multipliers. These Galois fieldmultipliers could be of a conventional design. In the alternative, theycould use the multiplier discussed with reference to FIGS. 13-17G. Theoutput of each multiplier 422 is provided on multiplier output 424 (i.e.424A to 424P) and is the product of the data stored in the memoryelement 412 (i.e. 412A to 412P) and the data stored in memory element416 (i.e. 416A to 416P). This stored data is provided to the multiplier422 by buses 414 (i.e. 414A to 414P) and 420 (i.e. 420A to 420P),respectively.

Each of the cells 402 includes a modulo two adder 426 (i.e. 426A to426P). The output of each adder 426 is provided on adder output 428(i.e. 428A to 428P) and is the sum of the data stored in memory element408 (i.e. 408A to 408P) and the data from multiplier output 424. Each ofthe cells includes a multiplexor 430 (i.e. 430A to 430P). Adder output428 is coupled to multiplexor input 454. The multiplexor 430 alternatelycouples its input 452 to its output 440 or its input 454 to its output440. The multiplexors described in this specification couple only oneinput at a time to their output. The multiplexor 430 is controlled bythe control circuit 406. This circuit 406 broadcasts first and secondcontrol signals to each of the multiplexors 430 using control line 434.In response to the first control signal, the multiplexor 430 couples itsinput 452 to its output 440. In response to the second control signal,the multiplexor 430 couples its input 454 to its output 440. In thepresent embodiment, the first control signal is a logic one and thesecond control signal is a logic zero. The input 452P of the multiplexor430P in the first cell 402P is coupled to the logic zero input 446. Theinput 452 in cells succeeding the first cell 402P (i.e. cells 402O to402A) is coupled to the output 418 of the memory element 416 in theimmediately preceding cell. In this specification, "preceding cell"shall refer to a cell that shifts data into another cell. "Succeedingcell" shall refer to a cell that receives data that has been shiftedfrom another cell. The output 440 of multiplexor 430 is coupled to theinput of memory element 416. The output of memory element 416 in eachpreceding cell 402 is coupled to the input 452 of the multiplexor 430 inthe immediately succeeding cell 402.

Received data input 448 is coupled to input 458 of multiplexor 436.Logic zero input 444 is coupled to input 460 of multiplexor 436.Multiplexor 436 is controlled by control circuit 404. In particular,control circuit 404 provides first and second control signals on controlline 442. In the present embodiment, the first control signal is a logicone, and the second control signal is a logic zero. In response to thefirst control signal, multiplexor 436 couples its input 460 to itsoutput 432. In response to the second control signal, multiplexor 436couples its input 458 to its output 432. The output 418A of the memoryelement 416A in the last cell 402A is coupled to the input 462 of themultiplexor 438. The input 464 of multiplexor 438 is coupled to thelogic zero input 456. Multiplexor 438 is controlled by control circuit406 and the first and second control signals on control line 434. Inresponse to the first control signal, multiplexor 438 couples its input462 to its output 446. In response to the second control signal,multiplexor 438 couples its input 464 to its output 446.

In operation, prior to receiving the received information stream r(x) oninput 448, the first control signals are applied to multiplexors 436,430 and 438. In this state, multiplexor 436 broadcasts a zero byte tothe input of each of memory elements 408. Memory elements 408 areclocked once to store this zero byte. Similarly, multiplexors 430 areused in a similar manner to latch zero bytes from the input 446 intomemory elements 416. In the present embodiment, memory elements 416 areclocked 16 times so that zero bytes are latched from input 446 into allof the memory elements 416. The input 462 of multiplexor 438 is coupledto its output.

After initializing syndrome calculator 304 in this manner, controlcircuits 404 and 406 provide the second control signal to multiplexors436, 430 and 438 to couple their inputs 458, 454 and 464 to theiroutputs 432, 440 and 446, respectively. In this state, calculator 304 isprepared to receive the information stream r(x).

The coefficients of r(x) are received at input 448 one byte at a time,starting with the highest degree coefficient r₂₀₃. The control circuit404 detects the presence of this coefficient and configures multiplexor436 to couple its input 458 to its output 432. Coefficient r₂₀₃ isbroadcast from input 448 onto bus 432 so that it is applied to the inputof each of the memory elements 408. At this point r₂₀₃ is clocked intomemory elements 408 by a first clock signal and the next coefficient ofr(x), r₂₀₂, is applied at the inputs of memory elements 408. At thispoint, because memory element 416 contains a zero byte, the output ofmultiplier 422 is zero. Accordingly, the output of adder 426 is r₂₀₃.This output is applied to the input of memory element 416 throughmultiplexor 430. The next clock cycle stores the coefficient r₂₀₂ inmemory elements 408 and stores r₂₀₃ in memory elements 416. After thissecond clock cycle, r₂₀₁ will be applied to the inputs of memoryelements 408. The output of multiplier 422 is r₂₀₃ α^(k) where k is thepower of a stored in the memory element 412 in the particular cell. Theoutput of the adder 426 is r₂₀₂ +r₂₀₃ α^(k). This output is appliedthrough multiplexor 430 to the input of memory element 416. The nextclock cycle stores r₂₀₁ in memory elements 408 and r₂₀₂ +r₂₀₃ α^(k) inmemory elements 416. The input to memory element 416, at this point isr₂₀₁ +r₂₀₂ α^(k) +r₂₀₃ α^(2k). The next clock cycle produces r₂₀₀ +r₂₀₁α^(k) +r₂₀₂ α^(2k) +r₂₀₃ α^(3k) at the input of memory elements 416.This process continues until all of the coefficients of r(x) have beenclocked into the syndrome calculator in this manner to provide in thememory elements 416 the coefficients ##EQU8## Thus, upon completion ofthis process, memory element 416A will contain syndrome S₀, memoryelement 416B will contain syndrome S₁ and memory elements 416C to 416Pwill contain the syndromes S₂ to S₁₅, respectively. At this point,control circuit 406 signals multiplexors 430 and 438 to create a serialpath of buses 418 and 440. The syndromes S₀ to S₁₅ are then shifted outof the syndrome calculator 304 in order through the output 446 ofmultiplexor 438. As described with respect to FIG. 3, these syndromesare passed to the error locator polynomial calculator 310 and to theerror transform calculator through the delay 320.

Error Locator Polynomial Calculator

FIG. 5 is a functional block diagram of the error locator polynomialcalculator 310. Other embodiments of the invention could use other errorlocator polynomial calculators. As shown in FIG. 5, this calculatorincludes 16 (2t) cells. Again, the number of cells can be varied tohandle RS codes that handle more or less errors, or if the embodimentuses cells to perform repeated calculations, for example. For ease ofillustration, cells 502C to 502O are shown symbolically. The cells 502Ato 502P perform the modified Euclid's algorithm to calculate the errorlocator polynomial. Each cell includes an R-Q degree calculator cell 512(i.e. cells 512A to 512P), an R-Q calculator cell 514 (i.e. cells 514Ato 514P), a trigger circuit cell 516 (i.e. cells 516A to 516P) and a λ-μcalculator cell 518 (i.e. cells 518A to 518P). Depending on the resultsof the calculation, the error locator polynomial will be output asλ_(final) or μ_(final) on bus lines 522 or 524, respectively. Thecontrol circuit 528 configures the multiplexor 526 to couple theappropriate bus line 522 or 524 to the output 530. The error locatorpolynomial λ(x) will be output at output 530 based upon thisconfiguration of multiplexor 526.

The following initial conditions are used by the present embodiment toimplement Euclid's iterative algorithm. ##EQU9##

In the present embodiment, the coefficients of each polynomial are 8 bitsymbols. Thus, the coefficient of the highest power of x in A(x) (i.e.X^(2t),) is 00000001. The coefficient of the lowest power of x μ₀ (x)(i.e.x⁰) is 00000001. These coefficients are generated by the syndromecalculator 304 at its output 446. The one bit can be provided in anyconvenient manner such by using a hardwired logic one or by generatingthe logic one using control circuitry. The coefficients of λ₀ (x) areprovided using a hardwired zero word. The deg (R₀ (x)) is hardwired tobe 16 (i.e. 2t) and the deg (Q₀(x)) is hardwired to be 15 (i.e. 2t-1).Leading zero coefficients of S(x) are not dropped when determining deg(Q₀ (x)).

The cells 502 of the error locator polynomial calculator 310 operate tocalculate R_(i) (x) from R_(i-1) (x) and Q_(i-1) (x), Q_(i) (x) fromR_(i-1) (x) or Q_(i-1) (x), λ_(i) (x) from λ_(i-1) (x) and μ_(i-1) (x),μ_(i) (x) from λ_(i-1) (x) or μ_(i-1) (x), deg(R_(i) (x)) fromdeg(R_(i-1) (x)) or deg (Q_(i-1) (x)) and deg(Q_(i) (x)) fromdeg(R_(i-1) (x)) or deg(Q_(i-1) (x)). The calculations performed onR_(i-1) (x) and Q_(i-1) (x) produce polynomials having lower degrees.Calculations are performed by cells 502 as long as the degree of one ofthe polynomials R_(i-1) (x) and Q_(i-1) (x) is greater than t.Initially, the degree of R_(i-1) (x) and Q_(i-1) (x) are greater than t.The same calculations that are performed on R_(i-1) (x) to obtain R_(i)(x) are performed on λ_(i-1) (x) to obtain λ_(i) (x). Similarly, thesame calculations that are performed on Q_(i-1) (x) to obtain Q_(i) (x)are performed on μ_(i-1) (x) to obtain μ_(i) (x). Once the degree ofeither R_(i-1) (x) or Q_(i-1) (x) is less than t, however, then theerror locator polynomial has been determined. In particular, if thedegree of R_(i-1) (x) is less than t, then the calculated λ_(i-1) (x) isthe error locator polynomial. If the degree of Q_(i-1) (x) is less thant and the degree of R_(i-1) (x) is not less than t, then the calculatedμ_(i-1) (x) is the error locator polynomial. Once the error locatorpolynomial has been calculated in this manner, no further calculationsare performed on λ_(i-1) (x) and μ_(i-1) (x). These final polynomials,which we shall label λ_(final) (x) and μ_(final) (x), are passedunmodified to the outputs 522 and 524, respectively. Similarly, thefinal values calculated by the degree calculator of deg(R_(final) (x))and deg(Q_(final) (x)) are passed out of the last cell of the calculator310. The values deg(R_(final) (x)) and deg(Q_(final) (x)) are the valuesof deg(R_(i) (x)) and deg(Q_(i) (x)) output from the cell 502 that alsofirst output the final polynomials λ_(final) (x) and μ_(final) (x). Onceλ_(final) (x) and μ_(final) (x) are output from the last λ₋μ calculatorcell 518P, the control circuit 528 sends control signals to themultiplexor 526 to configure it to pass one of these polynomials out ofoutput 530 as λ_(FINAL) (x). The control circuit 528 sends a firstcontrol signal to the multiplexor 526 to configure it to pass λ_(final)(x) out of output 530 as λ_(FINAL) (x) if deg(R_(final) (x)) is lessthan t. Otherwise, the control circuit 528 sends a second control signalto the multiplexor 526 to configure it to pass μ_(final) (x) out ofoutput 530 as λ_(FINAL) (x) if deg(R_(final) (x)) is not less than t.The coefficients of the error locator polynomial λ_(FINAL) (x) passedout of output 530 are passed to the input 1044 (FIG. 10) of the standarderror locator polynomial calculator 312.

Error Locator Polynomial Calculator--R-Q Calculator

FIG. 6 illustrates the R-Q calculator cell 514 which is used in theerror locator polynomial calculator 310. The R-Q calculator cell 514 ineach cell 502 calculates polynomials R_(i) (x) and Q_(i) (x) from thepolynomials R_(i-1) (x) and Q_(i-1) (x) using the following equations.

    R.sub.i (x)= σ.sub.i-1 b.sub.i-1 R.sub.i-1 (x)+σ.sub.i-1 a.sub.i-1 Q.sub.i-1 (x)!-x.sup.|1.sbsp.i-1.sup.|  σ.sub.i-1 a.sub.i-1 Q.sub.i-1 (x)+σ.sub.i-1 b.sub.i-1 R.sub.i-1 (x)!                                            (7)

    Q.sub.i (x)=σ.sub.i-1 Q.sub.i-1 (x)+σ.sub.i-1 R.sub.i-1 (x)(8)

where

    l.sub.i-1 =deg(R.sub.i-1 (x))-deg(Q.sub.i-1 (x))           (9)

    σ.sub.i-1 =1 if l.sub.i-1 ≧0                  (10)

    σ.sub.i-1 =0 if l.sub.i-1 <0                         (11)

and where a_(i-1) is the leading (highest degree) coefficient of R_(i-1)(x), and b_(i-1) is the leading coefficient of Q_(i-1) (x). The termsdeg(R_(i-1) (x)) and deg(Q_(i-1) (x)) represent the degree of thesepolynomials as calculated by the degree calculator cells 512. As we willdescribe, the degree calculated by cells 512 may not be the same as the"actual" degree of R_(i) (x) and Q_(i) (x). For example, the "actual"degree of 0x² +x+1 is typically considered to be 1. Under certaincircumstances, however, the degree calculator might treat the degree ofa polynomial such as 0x² +x+1 as 2.

When deg(R_(i-1) (x)) is greater than or equal to deg(Q_(i-1) (x)),these equations reduce to

    R.sub.i (x)=b.sub.i-1 R.sub.i-1 (x)-x.sup.|1.sbsp.i-1.sup.| a.sub.i-1 Q.sub.i-1 (x)(12)

    Q.sub.i (x)=Q.sub.i-1 (x)                                  (13)

When deg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)), these equationsreduce to

    R.sub.i (x)=a.sub.i-1 Q.sub.i-1 (x)-x.sup.|1.sbsp.i-1.sup.| b.sub.i-1 R.sub.i-1 (x)(14)

    Q.sub.i (x)=R.sub.i-1 (x)                                  (15)

Thus, the R-Q calculator cells 514 of the present embodiment implementequations (12) and (13) or equations (14) and (15) depending on therelative values of deg(R_(i-1) (x)) and deg(Q_(i-1) (x)) as calculatedby the degree calculator cells 512. The x.sup.|1.sbsp.i-1.sup.| inequations (12) and (14) simply represents the difference betweendeg(R_(i-1) (x)) and deg(Q_(i-1) (x)). By multiplying the lower degreepolynomial by x.sup.|1.sbsp.i-1.sup.|, the equations indicate that thesubtrahend and minuend of equations (12) and (14) have the same degree.By also cross multiplying R_(i-1) (x) by the leading coefficient ofQ_(i-1) (x) and Q_(i-1) (x) by the leading coefficient of R_(i-1) (x)and then subtracting in this manner, the coefficient of the term inR_(i) (x) having the same degree as the highest degree term of thesubtrahend and minuend will always be zero. As we will describe, thiszero coefficient of R_(i) (x) is dropped after the calculation ofequation (12) or (14) is performed. Dropping this zero coefficientreduces the degree of R_(i) (x) by one from R_(i-1) (x) (when equation(12) is used) or reduces the degree of R_(i) (x) by one from Q_(i-1) (x)(when equation (14) is used).

In operation, R_(i-1) (x) and Q_(i-1) (x) are input into each of therespective R-Q calculator cells 514A to 514P and R_(i) (x) and Q_(i) (x)are output from the R-Q calculator cell into which R_(i-1) (x) andQ_(i-1) (x) were input. Thus, looking at FIG. 5, for example, R₀ (x) andQ₀ (x) are input into calculator cell 514A on buses 508 and 510,respectively, and R₁ (x) and Q₁ (x) are output on these respectivebuses. Similarly, R₁ (x) and Q₁ (x) are input into calculator cell 514Bon buses 508 and 510, respectively, and R₂ (x) and Q₂ (x) are output.Calculators cells 514C to 514P operate similarly to produce R₃ (x)through R₁₆ (x) and Q₃ (x) through Q₁₆ (x).

The design of R-Q calculator cell 514 as shown in FIG. 6 enablesrepetitions of this same cell design to be used to implement calculatorcells 514A through 514P. Such a cell design makes the R-Q calculator ofthe present embodiment relatively easily scalable. In particular, thepresent R-Q calculator implemented by cells 514A through 514P can beadapted to provide the R-Q calculations for other Reed Solomon codes,for example, by simply increasing or decreasing the number of these R-Qcalculator cells 514 that are coupled in series.

Calculator cell 514 includes memory elements 602, 604, 606, 608, 618,620, 624 and 630 each having an input and an output. These memoryelements are all implemented as 8 bit registers, similar to registers408 and 416 of FIG. 4, which each provide the data that they are storingto their respective output. When each memory element is clocked by aclock signal (not shown), it stores the data that is present at itsinput. Calculator 514 also includes switch 610, R' multiplexor 632, Q'multiplexor 634, Galois field multipliers 622 and 628 and modulo twoadder 626. Control circuits 612 and 636 provide control signals tocontrol the switch 610 and the multiplexors 632 and 634 in calculator514. All of the bus lines in the calculator 514 are 8 bit wide buses,except control lines 646, 658 and 660. Control lines 646, 658 and 660are one bit lines.

The input of memory element 602 receives the coefficients of thepolynomial R_(i-1) (x), and the input of memory element 606 receives thecoefficients of the polynomial Q_(i-1) (x). The output of memoryelements 602 and 606 are coupled to the inputs of memory elements 604and 608, respectively. The outputs of memory elements 604 and 608 arecoupled to an R switch input 654 and a Q switch input 656, respectively,of switch 610. Control circuit 612 is coupled to switch 610 throughcontrol line 646.

Switch 610 has a n R' switch output 614 and a Q' switch output 616. Theoutput 614 is coupled to an R'(x) calculation path 640 and an R'(x)bypass path 638. The R'(x) bypass path 638 and the R'(x) calculationpath 640 are each coupled at their opposite ends to an input 662 andinput 664, respectively, of R' multiplexor 632. The output 616 of switch610 is coupled to a Q'(x) delay path 642 and a Q'(x) bypass path 644.The Q'(x) bypass path 644 and the Q'(x) delay path 642 are coupled attheir opposite ends to an input 668 and input 666, respectively, of Q'multiplexor 634. The output 650 of multiplexor 632 provides R_(i) (x)which might become, for example, the R_(i-1) (x) input of a succeedingR-Q calculator cell 514. Similarly, the output 652 of multiplexor 634provides Q_(i) (x) which might become, for example, the Q_(i-1) (x)input of a succeeding R-Q calculator cell 514. Control circuit 636 iscoupled to multiplexor 632 through control line 660 and to multiplexor634 through control line 658.

The R'(x) calculation path 640 includes the multiplier 622 coupled tothe adder 626. The R' switch output 614 provides a first input to themultiplier 622, and the output of memory element 620 provides a secondinput. The output of the multiplier 622 provides a first input to themodulo two adder 626. The second input to the adder 626 is provided by across couple path 648. Cross couple path 648 couples the output 616 ofswitch 610 to the input of the adder 626 through a multiplier 628. Thesecond input to the multiplier 628 is provided by the output of memoryelement 624. The output of multiplier 628 provides the second input toadder 626. The output of adder 626 is coupled to the input 664 ofmultiplexor 632.

The R'(x) bypass path 638 couples the R' output 614 of switch 610 to theinput 662 of multiplexor 632 through a memory element 618. The memoryelement 618 functions as a delay in this path 638.

The Q'(x) delay path 642 couples the output 616 of switch 610 to theinput 666 of multiplexor 634 through memory element 630. Similar tomemory element 618 in R'(x) bypass path 638, memory element 630functions as a delay in path 642. The Q'(x) bypass path couples theoutput 616 of switch 610 directly to the input 668 of multiplexor 634without a delay.

The control circuit 612 tests to determine if deg(R_(i-1) (x)) is lessthan deg(Q_(i-1) (x)) where deg (R_(i-1) (x)) and deg (Q_(i-1) (x)) arethe inputs to the degree calculator 512 in the same cell 502. If so,circuit 612 provides a first control signal to switch 610. If not, itprovides a second control signal to switch 610. In response to the firstcontrol signal, switch 610 passes the data present at its input 654 toits output 616 and passes the data present at its input 656 to itsoutput 614. In response to the second control signal, switch 610 passesthe data present at its input 654 to its output 614 and passes the datapresent at its input 656 to its output 616. We shall refer to the dataoutput from output 614 as R_(i-1) (x)' and the data output from output616 as Q_(i-1) (x)'. As this description illustrates R_(i-1) (x)' willeither be R_(i-1) (x) or Q_(i-1) (x), depending on whether or notdeg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)). Similarly, Q_(i-1) (x)'will either be R_(i-1) (x) or Q_(i-1) (x), depending on whether or notdeg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)).

The data stored in memory element 620 will either be a_(i-1), theleading coefficient of R_(i-1) (x), or b_(i-1), the leading coefficientof Q_(i-1) (x). This data will be a_(i-1) when deg(R_(i-1) (x)) is lessthan deg(Q_(i-1) (x)) and will be b_(i-1) when deg(R_(i-1) (x)) is notless than deg(Q_(i-1) (x)). The data stored in memory element 624 willalso be either a_(i-1) or b_(i-1). This data will be b_(i-1), whendeg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)) and will be a_(i-1) whendeg(R_(i-1) (x)) is not less than deg(Q_(i-1) (x)). Another way ofdescribing the data stored in memory elements 620 and 624 is that thedata in memory element 620 is b_(i-1), the leading coefficient ofQ_(i-1) (x)', and the data in memory element 624 is a_(i-1) ', theleading coefficient of R_(i-1) (x)'. Under certain circumstances thedecoder 202 might treat zero rather than one as the leading coefficientof a polynomial such as 0x² +x+1. Unless otherwise specified, "leadingcoefficient" shall refer to the coefficient of a polynomial that thedecoder 202 treats as the leading coefficient, whether or not thatcoefficient is zero.

If deg(R_(i-1) (x)) is greater than or equal to deg(Q_(i-1) (x)), thenR_(i-1) (x)'=R_(i-1) (x), Q_(i-1) (x)'=Q_(i-1) (x). As a result, thecalculator cell 514 implements equations (12) and (13). Memory element620 contains b_(i-1) ' (which in this case is b_(i-1)) and memoryelement 624 contains a_(i-1) ' (which in this case is a_(i-1)). Thesevalues provide the cross multiplication shown in equations (12) and(13). Because adder 626 is a modulo two adder, addition is the same assubtraction. Accordingly, adder 626 provides the subtraction shown inequations (12) and (13). In the present embodiment, the appropriatevalues of a_(i-1) and b_(i-1) are loaded into the memory elements 620and 624 by control circuit 670 which reads a_(i-1) and b_(i-1) from theoutput of memory elements 602 and 606 at nodes A and B. Control circuit670 is triggered by the start₋₋ eval₋₋ out signal from FIG. 5. It uses asignal from control circuit 612 to determine the appropriate one ofmemory elements 620 and 624 to which each of the coefficients issupplied.

If deg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)), then R_(i-1)(x)'=Q_(i-1) (x), Q_(i-1) (x)'=R_(i-1) (x). As a result, the calculatorcell 514 implements equations (14) and (15). In that situation, memoryelement 620 is loaded with b_(i-1) ' (which in this case is a_(i-1)) andmemory element 624 is loaded with a_(i-1) ' (which in this case isb_(i-1)). This loading occurs during the delays introduced by elements604 and 608 and again is accomplished using control circuit 670. Oncecontrol circuit 612 completes it's test, the appropriate coefficientsare routed to and clocked into memory elements 620 and 624.

In operation, after the syndrome polynomial S(x) is calculated, theleading coefficients of R_(i-1) (x) and Q_(i-1) (x) are clocked intomemory elements 602 and 606 at the same time. R_(i-1) (x) and Q_(i-1)(x) input into cell 514A are defined by the initial conditions (1) to(6). R_(i-1) (x) and Q_(i-1) (x) input into cells 514B to 514P aredefined by the output of the respective preceding cells. Clocking theleading coefficients in at the same time inherently performs themultiplication by x.sup.|1.sbsp.i-1.sup.| in equation (7). Inparticular, the leading coefficients are clocked in at the same time andoperated on by calculator 514 even if R_(i-1) (x) and Q_(i-1) (x) are ofdifferent degrees. This operation corresponds to multiplying the lowerdegree polynomial by x.sup.|1.sbsp.i-1.sup.|. During this first clockcycle, memory elements 620 and 624 are also being loaded with b_(i-1) 'and a_(i-1) ' as appropriate. Also during this first clock cycle, thecontrol circuit 612 determines which polynomial is of higher degree, andit provides the appropriate control signal to switch 610. Similarly,control circuit 636 determines whether or not the leading coefficient ofQ_(i-1) (x)' is zero, and it provides the appropriate control signals tomultiplexors 632 and 634. To do so, control circuit 636 must firstdetermine if Q_(i-1) (x)' will equal R_(i-1) (x) or Q_(i-1) (x) bydetermining if the degree of R_(i-1) (x) is less than the degree ofQ_(i-1) (x). Control circuit 636 relies upon control circuit 612 to makethis determination. Upon making this initial evaluation, control circuit636 determines if the leading coefficient of Q_(i-1) (x)' is zero.

The second clock cycle clocks the leading coefficients of R_(i-1) (x)and Q_(i-1) (x) into memory elements 604 and 608, respectively, and thecoefficients of R_(i-1) (x) and Q_(i-1) (x) that are one degree belowthese lead coefficients into memory elements 602 and 606, respectively.Assuming the leading coefficient of Q_(i-1) (x)' is not zero,multiplexors 632 and 634 couple to their respective outputs 650 and 652the R'(x) calculation path 640 and the Q'(x) delay path 642. The R'(x)calculation path performs the calculation of equation (12) or (14),depending on the configuration of switch 610. The leading coefficientoutput from path 640 is provided out of output 650. The delay introducedby memory element 630, however, causes the leading coefficient ofQ_(i-1) (x)' to be output out of output 652 one clock cycle after theleading coefficient from path 640 is output out of output 650. Inparticular, the lead coefficient of Q_(i-1) (x)' is provided at output652 at the same time as the second highest degree coefficient from path640 is provided at output 650. The lead coefficient out of output 650 isdropped because it is ignored by the succeeding R-Q calculator cell 514.It is ignored because it only took two clock cycles to pass this leadcoefficient to output 650. The trigger circuit 516, discussed below,does not trigger the succeeding cell 502 until 3 clock cycles havepassed. Accordingly, in this circumstance the trigger circuit 516 causesthe succeeding cell 502 to clock in the second coefficient out of path640 and the lead coefficient of Q_(i-1) (x)'.

If the leading coefficient of Q_(i-1) (x)' is zero, multiplexors 632 and634 will pass to their respective outputs 650 and 652 the data from theR(x) bypass path 638 and the data from the Q(x) bypass path 644. Bydoing so, R_(i) (x) will be R_(i-1) (x)'. In this case, the leading zerocoefficient of Q_(i-1) (x)' is being dropped. By bypassing delay 630,the lead coefficient of Q_(i-1) (x)' is passed out of output 652 in twoclock cycles. Accordingly, it is dropped in the same manner as the leadcoefficient of R_(i-1) (x)' when the lead coefficient of Q_(i-1) (x)' isnot zero. When the lead coefficient of Q_(i-1) (x)' is zero, equations(12)-(15) are not implemented. Rather, the coefficients are passedwithout calculation as shown in FIG. 6.

R-Q calculator 512 only drops at most one zero coefficient of R_(i) (x)or Q_(i) (x) as appropriate. Accordingly, zero coefficients of R_(i)(x), and Q_(i) (x) having a degree below the dropped zero coefficient ofeach of these polynomials are not dropped. Such zero coefficients mustbe clocked into the memory elements of the calculation circuits tomaintain proper timing to provide the proper calculations. As we willdescribe, this feature has enabled the R-Q degree calculators to bedesigned so that they reduce the degree of deg (R_(i-1) (x)) or deg(Q_(i-1) (x)) by at most one.

Error Locator Polynomial Calculator--Trigger Circuit

FIG. 7 illustrates the trigger circuit cell 516 used by the presentembodiment. The trigger circuit cell 516 of each preceding cell 502signals each succeeding cell 502 to indicate that valid data is presentat the inputs of the succeeding cell and that the succeeding cell shouldstart the calculations using that data. Accordingly, for example, thetrigger circuit cell 516A signals cell 502B to begin calculations;trigger circuit cell 516B signals cell 502C to begin calculations and soon. The trigger signal is applied to each cell 502 using a one bittrigger line 520. The trigger circuit 516A of the first cell 502A istriggered by a controller of the syndrome calculator (not shown) afterthe syndrome polynomial has been calculated by calculator 304. Thetrigger circuit output "start₋₋ eval₋₋ out" of the last cell 502Psignals the standard error locator polynomial calculator 312 to startcalculating the standard error locator polynomial from the coefficientsprovided out of output 530 (FIG. 5). Start₋₋ eval₋₋ out of FIG. 7 iscoupled to "start₋₋ calc₋₋ in" of FIG. 10.

As illustrated in FIG. 7, the cell 516 provides an adaptable delay oftwo clocks cycles or three clock cycles. In particular, this cellincludes memory elements 702, 704 and 706. These memory elements aresimilar to the memory elements described previously in that they provideto their outputs the data they store. When these memory elements areclocked by a clock signal (not shown), they store the data present attheir respective inputs. Rather than being 8 bit registers, however,these memory elements are single bit registers.

The input of memory element 702 provides the input of trigger circuitcell 516. The output of memory element 702 is coupled to the input ofmemory element 704. The output of memory element 704 is coupled tomultiplexor 708 either through bypass path 710 or through delay path718. Bypass path 710 couples the output of memory element 704 to a firstinput of multiplexor 708 without delay. Delay path 718 couples theoutput of memory element 704 to the input of memory element 706 and theoutput of memory element 706 to a second input of multiplexor 708. Theoutput 712 of the multiplexor 708 provides the output of the triggercircuit cell 516. The trigger circuit cell requires two cycles to clocka trigger signal (e.g. a high logic level) from the input 716 to theoutput 712 when the bypass path 710 is activated. The trigger circuitcell requires three cycles to clock a trigger signal from the input 716to the output 712 when the delay path 718 is activated. In particular,the delay path 718 introduces a delay of one additional clock cycle.

The control circuit 714 determines if deg (R_(i-1) (x)) or deg (Q_(i-1)(x)) coming into a particular cell 502 is less than t. If the degree ofneither of these polynomials is less than t, the delay produced bytrigger circuit 516 will be three clock cycles. This three clock cycledelay is the delay that drops coefficients in the manner we havedescribed. When the leading zero coefficient of R(x) is dropped, thedegree of R_(i) (x) is reduced by one from the polynomial R_(i-1) (x)'.When the leading zero coefficient of Q(x) is dropped, the degree ofQ_(i) (x) is reduced by one from Q_(i-1) (x)'.

If the degree of either of these polynomials is less than t, the delayproduced by trigger circuit 516 will be two clock cycles. As we haveexplained above, in the initial R-Q calculator cells of the calculator310 illustrated in FIG. 5, such as cell 502A, the degree of both R_(i-1)(x) and Q_(i-1) (x) will be greater than t. A three clock cycle delay inthe trigger circuit is necessary for proper operation of the R-Qcalculators under these circumstances. As we will explain below,however, when the degree of either R_(i-1) (x) or Q_(i-1) (x) dropsbelow t, the λ-μ calculator cells 518 pass the λ(x) polynomial and theμ(x) polynomial without subjecting them to further calculations.Accordingly, the delay of the trigger circuit cells 516 is shortened tomatch the delay of the λ-μ calculator cells when they are in this modeof operation. The trigger circuit cells 516 must trigger the standarderror locator polynomial calculator 312 at the appropriate time to clockin the coefficients of the error locator polynomial λ(x).

This timing between the trigger circuit and the R-Q calculator cellsensures that the degree of R_(i-1) (x) or Q_(i-1) (x) will be reduced byone, but by no more than one in each cell 502. For example, if thecalculation of equations (12) and (13) produces the polynomial 0x³ +0x²+x+1, the degree will only be reduced by one, and the polynomial 0x²+x+1 will be passed to the subsequent cell. In particular, the triggercircuit timing is such that the 0x³ term is dropped. The 0 coefficientof the x² term would be clocked into the subsequent R-Q calculator,however, just as a non-zero coefficient of x² would be. This feature ofthe present embodiment simplifies the design of the degree calculatorcells 512.

The correspondence between the timing of the trigger circuit of FIG. 7and of the R-Q calculator of FIG. 6 can be illustrated as follows.Assume Q_(i-1) (x)' is not zero. In this situation, the leadingcoefficients of R_(i-1) (x) and Q_(i-1) (x) are clocked into the memoryelements 602 and 606, respectively, by a first clock cycle. This clockcycle corresponds to the trigger bit being clocked into memory element702 of the FIG. 7 trigger circuit cell. The leading coefficients ofR_(i-1) (x) and Q_(i-1) (x) are clocked into the memory element 604 and608, respectively, by a second clock cycle. This clock cycle correspondsto the trigger bit being clocked into memory element 704 of the FIG. 7trigger circuit cell. Once the leading coefficients of R_(i-1) (x) andQ_(i-1) (x) are clocked into memory elements 604 and 608, the leadingzero coefficient produced by calculation path 640 is applied out ofoutput 650 to the input of the R-Q calculator in the succeeding cell502. The memory element 630 prevents the leading coefficient of Q_(i-1)(x)' from being applied out of output 652 to the input of the R-Qcalculator in the succeeding cell 502 by this second clock cycle. Athird clock cycle, however, will pass the leading coefficient of Q_(i-1)(x)' out of output 652 to the input of the memory element 606 of the R-Qcalculator in the succeeding cell 502. This same third clock cycle willclock the second coefficient of R_(i-1) (x) and Q_(i-1) (x) into memoryelements 604 and 608 so that the appropriate second coefficient isapplied out of output 650 to the input of memory element 602 of the R-Qcalculator in the succeeding cell 502. This third clock cycle will alsocause the trigger bit to be clocked into the memory element 706 of thetrigger circuit cell of FIG. 7 and out of output 712. The trigger bitpassed out of the output 712 triggers the succeeding cell 502.Accordingly, the leading coefficient of Q_(i-1) (x)' and the secondcoefficient produced by calculation path 640 will be latched into thememory elements 606 and 602, respectively, of the succeeding R-Qcalculator cell 514. The leading zero coefficient produced bycalculation path 640 was dropped because it was not latched into thesucceeding cell 514.

Error Locator Polynomial Calculator--R-Q Degree Calculator

FIG. 8 illustrates an R-Q degree calculator cell 512. The R-Q degreecalculator cells 512 calculate the degree of the polynomials R_(i) (x)and Q_(i) (x) from the degree of the polynomials R_(i-1) (x) and Q_(i-1)(x). Accordingly, the degree of R₀ (x) is input into the degreecalculator cell 512A on bus 504 and the degree of R₁ (x) is output.Similarly, the degree of Q₀ (x) is input into the degree calculator cell512A on bus 506 and the degree of Q₁ (x) is output. The degree of R_(i)(x) and Q_(i) (x) will be less than or equal to the degree of R_(i-1)(x) and Q_(i-1) (x), respectively. The degree calculator cells 512B to512P in cells 502B to 502P operate similarly. As demonstrated by theinitial conditions, (1) to (6), in the present embodiment the degree ofR₀ (x) is 16 (2t), and the degree of Q₀ (x) is 15 (2t-1). Because themaximum degree is 16, the bus lines 504 and 506 in FIG. 5 are five bitbuses.

Cell 512 includes memory elements 802 and 804, switch 810, multiplexors816, 818, 840 and 842, R(x) reduction path 820, R(x) reduction bypasspath 824, Q(x) reduction path 822 and Q(x) reduction bypass path 826.The R(x) reduction path 820 and the R(x) reduction bypass path 824 shallcollectively be referred to as the R(x) degree processing path, and theQ(x) reduction path 822 and the Q(x) reduction bypass path 826 shallcollectively be referred to as the Q(x) degree processing path. The cell512 also includes the R(x) degree processing bypass path 848 and theQ(x) degree processing bypass path 850. The memory elements 802 and 804are implemented as 5 bit registers each of which provide the data thatthey are storing to their respective output. When these memory elementsare clocked by a clock signal (not shown), they store the data presentat their respective inputs. Control circuits 612, 636 and 714 providecontrol signals to control the switch 810 and the multiplexors 816, 818,840 and 842. All of the buses of degree calculator cell 512 exceptcontrol lines 859, 860 and 862 are 5 bit paths. Control lines 859, 860and 862 are single bit lines. Other embodiments can use other buswidths, as appropriate.

The input of memory elements 802 and 804 provide the inputs 872 and 874,respectively. The output of memory element 802 is coupled to an input828 of switch 810 and to degree processing bypass path 848. Degreeprocessing bypass path 848 is coupled to an input 864 of multiplexor840. The output of memory element 804 is coupled to an input 830 ofswitch 810 and to degree processing bypass path 850. Degree processingbypass path 850 is coupled to an input 870 of multiplexor 842.

Switch 810 operates in the same manner as switch 610 in FIG. 6. Inparticular, degree calculator 512 uses the same control circuit 612 thatwas used by R-Q calculator 514 to determine if the deg (R_(i-1) (x)) isless than deg (Q_(i-1) (x)). If so, the control circuit 612 provides afirst control signal to switch 810. If not, it provides a second controlsignal to switch 810. In response to the first control signal, switch810 couples its input 828 to its output 834 and couples its input 830 toits output 832. In response to the second control signal, switch 810couples its input 828 to its output 832 and couples its input 830 to itsoutput 834.

The output 832 of switch 810 is coupled to the R(x) degree processingpath. In particular, output 832 is coupled to the R(x) reduction path820 and to the R(x) reduction bypass path 824. The R(x) reduction bypasspath 824 is coupled to an input 852 of multiplexor 816, and the R(x)reduction path 820 is coupled to an input 854 of multiplexor 816.Similarly, the output 834 of switch 810 is coupled to the Q(x) degreeprocessing path. In particular, output 834 is coupled to the Q(x)reduction path 822 and to the Q(x) reduction bypass path 826. The Q(x)reduction path 822 is coupled to an input 856 of multiplexor 818, andthe Q(x) reduction bypass path 826 is coupled to an input 858 ofmultiplexor 818. Multiplexors 816 and 818 are controlled by the samecontrol circuit 636 that was used by the R-Q calculator cell 514 of FIG.6. This control circuit 636 provides a first control signal to thesemultiplexors if the lead coefficient of Q_(i-1) (x)' is zero and asecond control signal to these multiplexors if the lead coefficient ofQ_(i-1) (x)' is not zero. In response to the first control signal,multiplexors 816 and 818 couple their inputs 852 and 856 to theiroutputs 836 and 838, respectively. In response to the second controlsignal, multiplexors 816 and 818 couple their inputs 854 and 858 totheir outputs 836 and 838, respectively. The output 836 of themultiplexor 816 is coupled to the input 866 of multiplexor 840. Theoutput 838 of the multiplexor 818 is coupled to the input 868 of themultiplexor 842.

Control circuit 714 provides to multiplexors 840 and 842 a first controlsignal if either of deg (R_(i-1) (x)) or deg (Q_(i-1) (x)) is less thant. Control circuit 714 provides to multiplexors 840 and 842 a secondcontrol signal if neither of deg (R_(i-1) (x)) or deg (Q_(i-1) (x)) isless than t. In response to the first control signal, multiplexors 840and 842 couple their inputs 864 and 870 to their outputs 844 and 846,respectively. In response to the second control signal, multiplexors 840and 842 couple their inputs 866 and 868 to their outputs 844 and 846,respectively.

As we have noted, the deg (R_(i-1) (x)) and deg (Q_(i-1) (x)) can bereduced by at most one in the R-Q calculator cell 514. Accordingly, thedegree calculator need only reduce the degree of the appropriatepolynomial by one. While the R(x) degree reduction path is shown toinclude an adder 812 and a stored value of -1, a decrementer may be usedto accomplish this operation. Alternatively, embodiments of theinvention may use a binary adder (rather than a modulo 2 adder) toimplement this function. In such an embodiment, the -1 value can bestored in a 5 bit register similar to those used elsewhere in thisinvention. Other storage techniques, such as some kind of ROM could beused. The output 832 of switch 810 provides a first input to the adderand the stored -1 data provides the second input to the adder. Theoutput of the adder will provide the degree applied to the first adderinput reduced by one. The Q(x) degree reduction path operates similarly.

In operation, deg (R_(i-1) (x)) is applied at input 872, and deg(Q_(i-1) (x)) is applied at input 874. Deg (R_(i-1) (x)) and deg(Q_(i-1) (x)), each a 5 bit word, are known from the output of apreceding degree calculator cell 512 or from the initial conditions.When clocking this data into memory elements 802 and 804, switch 810 andmultiplexors 816, 818, 840 and 842 are configured.

Control circuit 612 compares deg (R_(i-1) (x)) and deg (Q_(i-1) (x)). Ifdeg (R_(i-1) (x))<deg (Q_(i-1) (x)) then the first control signal isapplied to switch 810, coupling input 828 to output 834 and input 830 tooutput 832. If deg (R_(i-1) (x)) is not less than deg (Q_(i-1) (x)),then the second control signal is applied to switch 810, coupling input828 to output 832 and input 830 to output 834. The data output at output832 shall be called deg (R_(i-1) (x))'. The data output at output 834shall be called deg (Q_(i-1) (x))'.

Control circuit 636 determines if the lead coefficient of Q_(i-1) (x)'in the same cell 502 is equal to 0. If this lead coefficient is zero,the first control signals are applied to multiplexors 816 and 818,coupling the input 852 to the output 836 and the input 856 to the output838. If the lead coefficient is not zero, the second control signals areapplied to multiplexors 816 and 818, coupling the input 854 to theoutput 836 and the input 858 to the output 838.

Control circuit 714 determines if deg (R_(i-1) (x))<t or if deg (Q_(i-1)(x))<t. If so, then the first control signals from circuit 714 areapplied to multiplexors 840 and 842, configuring them as we havedescribed. If neither of deg (R_(i-1) (x))<t and deg (Q_(i-1) (x))<t,then the second control signals from circuit 714 are applied tomultiplexors 840 and 842, configuring them as we have described.

The calculator 512 is configured in this manner during a first clockcycle. Also, the first clock cycle clocks deg (R_(i-1) (x)) into memoryelement 802 and clocks deg (Q_(i-1) (x)) into memory element 804. Deg(R_(i) (x)) is output from output 844 and deg (Q_(i) (x)) is output fromoutput 846. Calculator 512 passes the data according to theconfigurations of switch 810 and multiplexors 816, 818, 840 and 842.

Multiplexors 816 and 818 cause the degree reduction accomplished by cell512 to track the degree reduction accomplished by the R-Q calculatorcell 514 in the same cell 502. Thus, for example, if the leadingcoefficient of Q_(i-1) (x)' is not zero, then the R-Q degree calculatoruses path 820 to reduce the deg (R_(i-1) (x))' by one and uses bypasspath 826 to not reduce the deg (Q_(i-1) (x))'. If the leadingcoefficient of Q_(i-1) (x)' is zero, on the other hand, then the R-Qcalculator uses the path 822 to reduce deg (Q_(i-1) (x))' by one anduses bypass path 826 to not reduce deg (R_(i-1) (x))'. The degreereduction calculator cell 512 accomplishes its calculations and providesthe results to its outputs 844 and 846 in one clock cycle. The resultsare provided at these outputs to be clocked into the subsequent cell502.

Processing bypass paths 848 and 850 paths are used when deg (R_(i-1)(x)) or deg (Q_(i-1) (x)) is less than t. Under such circumstances, thedegree calculator does not further reduce deg (R_(i-1) (x)) and deg(Q_(i-1) (x)), but passes them unchanged to the outputs of the lastdegree calculator cell 51 2P as deg (R_(final)) and deg (Q_(final)). Thevalue of deg (R_(final) is used by control circuit 528 in FIG. 5 toconfigure multiplexor 526 as we have described.

Error Locator Polynomial Calculator--λ-μ Calculator

FIG. 9 illustrates a λ-μ calculator cell 518. The λ-μ calculator cell518 in each cell 502 calculates the polynomials λ_(i) (x) and μ_(i) (x)from the polynomials λ_(i-1) (x) and μ_(i-1) (x) using the followingequations.

    λ.sub.i (x)= σ.sub.i-1 λ.sub.i-1 (x)+σ.sub.i-1 a.sub.i-1 μ.sub.i-1 (x)!-(x).sup.i-1  σ.sub.i-1 a.sub.i-1μ.sub.i-1 (x)+σ.sub.i-1 b.sub.i-1 λ.sub.i-1 (x)!(16)

    μ.sub.i (x)=σ.sub.i-1 μ.sub.i-1 (x)+σ.sub.i-1 λ.sub.i-1 (x)                                      (17)

where a_(i-1) and b_(i-1) are still the leading non-zero coefficients ofR_(i-1) (x) and Q_(i-1) (x), respectively. As these equationsdemonstrate, λ_(i) (x) is calculated in the same manner as R_(i) (x),and μ_(i) (x) is calculated in the same manner as Q_(i) (x).Accordingly, when the degree of R_(i-1) (x) is greater than or equal tothe degree of Q_(i-1) (x), these equations reduce to

    λ.sub.i (x)=b.sub.i-1 λ.sub.i-1 (x)-x.sup.i-1 a.sub.i-1 μ.sub.i-1 (x)                                          (18)

    μ.sub.i (x)=μ.sub.i-1 (x)                            (19)

When the degree of R_(i-1) (x) is less than the degree of Q_(i-1) (x),these equations reduce to

    λ.sub.i (x)=a.sub.i-1 μ.sub.i-1 (x)-x.sup.i-1 b.sub.i-1 λ.sub.i-1 (x)                                      (20)

    μ.sub.i (x)=λ.sub.i-1 (x)                        (21)

Cells 518 implement the reduced equations (18) and (19) or (20) and(21), again, depending on the relative values of deg(R_(i-1) (x)) anddeg(Q_(i-1) (x)). With reference to FIG. 5, λ_(i-1) (x) and μ_(i-1) (x)are input into each of the respective λ-μ calculator cells 518A to 518Pon bus lines 522 and 524, respectively, and λ_(i) (x) and μ_(i) (x) areoutput on bus lines 522 and 524, respectively, from the λ-μ calculatorcell into which λ₁₋₁ (x) and μ_(i-1) (x) were input. Thus, for example,λ₀ (x) and μ₀ (x) are input into calculator cell 518A on buses 522 and524, respectively, calculations are performed and λ_(i) (x) and μ_(i)(x) are output on the respective buses 522 and 524. Similarly, λ_(i) (x)and μ_(i) (x) are input into calculator cell 518B on buses 522 and 524,respectively, calculations are performed and λ₂ (x) and μ₂ (x) areoutput on the buses 522 and 524. Cells 518C to 518P operate similarly toproduce λ₃ (X) through λ₁₅ (x) and μ₃ (X) to μ₁₅ (X).

With reference to FIG. 9, cell 518 operates in basically the same manneras the R-Q calculator cell 514. Accordingly, the description of cell 518focuses on the differences. Elements numbered in FIG. 9 with a 900series number operate in a similar manner to the corresponding 600series element of FIG. 6.

Similar to calculator 514, λ_(i-1) (x) is clocked through memoryelements 902 and 904, and μ_(i-1) (x) is clocked through memory elements906 and 908. Switch 910 operates in the same manner as switch 610 and iscontrolled by the same control circuit 612. The λ'(x) calculation path940 and the λ'(x) bypass path 938 (collectively referred to as the λ(x)processing path) operate in a similar manner to the R'(x) calculationpath 640 and the R'(x) bypass path 638. The μ'(x) delay path 942 and theμ'(x) bypass path 944 (collectively referred to as the μ(x) processingpath) operate in a similar manner to the Q'(x) delay path 642 and theQ'(x) bypass path 644, respectively. The control circuit 936 providescontrol signals to the λ' multiplexor 932 and to the μ' multiplexor 934.Control circuit 936 operates differently than control circuit 636. Inparticular, control circuit 936, in addition to testing to determine ifthe leading coefficient of Q_(i-1) (x)' is zero, also tests to determineif deg (Q_(i) (x)) is greater than or equal to t. Deg (Q_(i) (x)) is thedegree of Q_(i) (x) as output from output 652 of the degree calculator512 in the same cell 502 as the present λ-μ calculator (e.g., the λ-μcalculator in cell 502A uses the output deg (Q_(i) (x)) from the degreecalculator in cell 502A). Due to the lower number of clock cyclesrequired by the degree calculator cell 512, the cell 512 will havecalculated deg (Q_(i) (x)) in time for control circuit 936 to test itand to configure the multiplexors 932 and 934. This additional conditiontested by control circuit 936 operates such that if either the leadcoefficient of Q_(i-1) (x)' is not zero or deg (Q_(i) (x))<t, then thelead coefficient of λ_(i-1) (x)' will be dropped rather than the leadcoefficient of μ_(i-1) (x)'. In particular, λ' multiplexor 932 will passthe data from λ(x) processing path 940, and λ' multiplexor 934 will passthe data from μ'(x) delay path 942. The delays 918 and 930 create timingin conjunction with trigger circuit 516 in the same manner as was donein R-Q calculator 514, except the delay adjustment circuit, whichincludes delay element 974, delay bypass path 971 and delay adjustmentmultiplexor 980 prevent the lead coefficient of λ_(i-1) (x)' from beingdropped under some circumstances. In particular, the delay element 974operates to prevent the lead coefficient of λ_(i-1) (x)' from beingdropped when deg (R_(i) (x))<t. When deg(R_(i) (x)) (i.e. the degreejust calculated in an R-Q degree calculator cell of a particular cell502) is less than t, then the lead coefficient λ_(i-1) (x)' should notbe dropped in the λ-μ calculator of that same cell 502. Accordingly,control circuit 997 provides a control signal based upon the comparisonof deg (R_(i) (x)) and t to cause the data to travel through delayelement 974. As a result of the insertion of this delay element, thelead coefficients of the polynomials calculated by the λ-μ calculatorwill be provided out of outputs 962 and 964 during the same clock cycle.Accordingly, no coefficients of either polynomial will be dropped.

The λ processing bypass path 966 is coupled between the λ switch input954 of switch 910 and input 984 to λ bypass multiplexor 958. The μprocessing bypass path 968 is coupled between the μ switch input 956 ofswitch 910 and input 990 to the μ bypass multiplexor 960. The input 982of λ bypass multiplexor 958 is coupled to the output of delay adjustmentmultiplexor 980. The input 986 of μ bypass multiplexor 960 is coupled tothe output 952 of μ' multiplexor 934. λ and μ processing bypass paths966 and 968 are used when deg (R_(i-1) (x)) or deg (Q_(i-1) (x)) is lessthan t. Under such circumstances, the λ-μ calculator cell 518 does notperform any additional calculations on λ_(i-1) (x) and μ_(i-1) (x).Under such circumstances, λ_(i-1) (x) and μ_(i-1) (x) are λ_(final) (x)and μ_(final) (x). Accordingly, they are passed to the inputs ofmultiplexor 526 (of FIG. 5) by the λ and μ bypass paths 966 and 968 inany cells 502 succeeding the calculation of λ_(final) (x) and μ_(final)(x). As described, either λ_(final) (x) or μ_(final) (x) is the errorlocator polynomial λ_(FINAL) (x). Accordingly, no additionalcalculations are performed on them because any additional calculationswould corrupt the calculated error locator polynomial. Bypass paths suchas paths 966 and 968 were not needed in the R-Q calculator 514 becauseonce the error locator polynomial has been determined, the presentembodiment no longer uses the polynomials calculated by the R-Qcalculator. Accordingly, it does not matter if the R-Q calculatorcontinues to perform calculations. The λ and μ bypass paths 966 and 968are activated by control signals provided to the λ and μ bypassmultiplexor's 958 and 960 by the control circuit 714. This controlcircuit provides a first control signal to each of multiplexor's 958 and960 when one of deg (R_(i-1) (x)) and deg (Q_(i-1) (x)) is less than t.It provides a second control signal to each of multiplexor's 958 and 960when neither of deg (R_(i-1) (x)) and deg (Q_(i-1) (x)) is less than t.When the first control signal is provided, the inputs 984 and 990 ofmultiplexors 958 and 960 are coupled to their respective outputs 962 and964. When the second control signal is provided, the inputs 982 and 986of multiplexors 958 and 960 are coupled to their respective outputs 962and 964. All of the buses in calculator 518 are 8 bits wide exceptcontrol lines 946, 958, 960, 992, 994 and 996. These control lines areone bit lines.

Similar to the R-Q calculator, the data stored in memory element 920will either be a_(i-1), the leading coefficient of R_(i-1) (x), orb_(i-1), the leading coefficient of Q_(i-1) (x). This data will bea_(i-1), when deg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)) and will beb_(i-1) when deg(R_(i-1) (x)) is not less than deg(Q_(i-1) (x)). Thedata stored in memory element 924 will also be either a_(i-1) orb_(i-1). This data will be b_(i-1) when deg(R_(i-1) (x)) is less thandeg(Q_(i-1) (x)) and will be a_(i-1) when deg(R_(i-1) (x)) is not lessthan deg(Q_(i-1) (x)). Another way of describing the data stored inmemory elements 920 and 924 is that the data in memory element 920 isb_(i-1), the leading coefficient of Q_(i-1) (x)', and the data in memoryelement 924 is a_(i-1) ', the leading coefficient of R_(i-1) (x)'.

If deg(R_(i-1) (x)) is greater than or equal to deg(Q_(i-1) (x)), thenλ_(i-1) (x)'=λ_(i-1) (x), μ_(i-1) (x)'=μ_(i-1) (x). As a result, thecalculator cell 518 implements equations (18) and (19). Memory element920 contains b_(i-1) ' (which in the case of is b_(i-1)) and memoryelement 924 contains a_(i-1) ' (which in this case is a_(i-1)). Thesevalues provide the cross multiplication shown in equations (18) and(19). Because adder 926 is a modulo two adder, addition is the same assubtraction. Accordingly, adder 926 provides the subtraction shown inequations (18) and (19). In the present embodiment, the appropriatevalues of a_(i-1) and b_(i-1) are loaded into the memory elements 920and 924 by control circuit 970 which reads a_(i-1) and b_(i-1) from theoutput of memory elements 902 and 906. Control circuit 970 is triggeredby the start₋₋ eval₋₋ out signal from FIG. 5. It uses a signal fromcontrol circuit 612 to determine the appropriate one of memory elements920 and 924 to which each of the coefficients is supplied.

If deg(R_(i-1) (x)) is less than deg(Q_(i-1) (x)), then λ_(i-1)(x)'=μ_(i-1) (x), μ_(i-1) (x)'=λ_(i-1) (x). As a result, the calculatorcell 518 implements equations (20) and (21). In that situation, memoryelement 920 is loaded with b_(i-1) (which in this case is a_(i-1)) andmemory element 924 is loaded with a_(i-1) ' (which in this case isb_(i-1)). This loading occurs during the delays introduced by elements904 and 908 and again is accomplished by control circuit 970. Oncecontrol circuit 612 completes it's test, the appropriate coefficientsare routed to and clocked into memory elements 920 and 924. Thesecoefficients are obtained from nodes A and B shown in FIG. 6.

Standard Error Locator Polynomial Calculator

FIG. 10 illustrates a standard error locator polynomial calculatorcircuit 312 that is used by the present embodiment. This calculatorcircuit 312 includes a trigger circuit 1002 and a standard polynomialcalculator 1004. As we have noted, the standard error locator polynomialcalculator 312 calculates σ(x) from λ(x).

    σ(x)=σ.sub.0 x.sup.8 +σ.sub.1 x.sup.7 +σ.sub.2 x.sup.6 +. . . σ.sub.8

where σ_(k) =λ_(k) /λ_(lnz) 0≦k≦t where σ_(k) represents thecoefficients of σ(x), λ_(lnz) is the highest order non-zero coefficientof λ(x) and λ(x) is the error locator polynomial calculated by thecalculator 310 and output on output 530 in FIG. 5. Accordingly, thecircuit 312 divides the error locator polynomial λ(x) by its leading(highest degree) non-zero coefficient.

The trigger circuit 1002 includes a memory element 1006 and a memoryelement 1008. These memory elements are single bit registers thatprovide to their output the data that they store. When these memoryelements are clocked by a clock signal (not shown), they store the datapresent at their respective inputs. These memory elements are operatingas delays. The input 1010 of the memory element 1006 provides the inputof the trigger circuit 1002. This input 1010 is coupled to the "start₋₋eval₋₋ out" output shown in FIG. 5. The output of the memory element1006 is coupled to the input of the memory element 1008. The output 1012of the memory element 1008 provides the output of the trigger circuit.The trigger circuit 1002 does not affect the operation of the standardpolynomial calculator. It passes the trigger signal to the errortransform calculator 314. The bus lines in the trigger circuit 1002 aresingle bit lines.

The standard polynomial calculator 1004 includes a multiplexor 1014having an input 1044 and an input 1042. The input 1044 is coupled to theoutput 530 of the error locator polynomial calculator 310 of FIG. 5.This input 1044 receives the coefficients of the error locatorpolynomial as they are provided at the output 530 of the error locatorpolynomial calculator 310. The input 1042 is coupled to a zero logiclevel. The multiplexor 1014 is controlled by the control circuit 1040.This circuit 1040 provides first and second control signals to themultiplexor 1014 in response to a trigger signal applied at the input1010. In response to the first control signal, the multiplexor 1014couples its input 1044 to its output 1046 and isolates its input 1042from its output 1046. In response to the second control signal, themultiplexor 1014 couples its input 1042 to its output 1046. The secondcontrol signal is applied to the multiplexor 1014 until the triggersignal is received from start₋₋ eval₋₋ out line of the error locatorpolynomial calculator 310. At this point, the control circuit 1040applies the first control signal to the multiplexor 1014.

The output 1046 of the multiplexor 1014 is coupled to the input of amemory element 1016. The output of memory element 1016 is coupled to theinput of memory element 1018. The output of memory element 1018 iscoupled to an input 1052 of multiplier 1020. Multiplier 1020 is a Galoisfield multiplier.

Inverter path 1056 is coupled between the output 1046 of multiplexor1014 and the input 1054 of multiplier 1020. Inverter path 1056 is formedas follows. The output 1046 is coupled to an input 1048 of multiplexor1024. The output of the multiplexor 1024 is coupled to an input of amemory element 1022. The memory elements 1016, 1018 and 1022 are 8 bitregisters that provide at their outputs the data that they are storing.When each of these memory elements is clocked by a clock signal (notshown), they store the data present at their respective inputs. Theoutput of the memory element 1022 is coupled to an input of a memoryelement 1030. The output of memory element 1022 is also coupled byfeedback path 1028 to the input 1050 of multiplexor 1024.

Memory element 1030 is a 256 byte ROM. This ROM receives at its addressinput 1048 an 8 bit binary word. The 8 bits of data stored in the ROM atthe address represented by this 8 bit word are the reciprocal (usingGalois field arithmetic) of the 8 bit word. Accordingly, the dataprovided at the output 1051 is the reciprocal of the data provided atthe input 1048. In the present embodiment, when a zero byte is presentat the input of ROM 1030, ROM 1030 outputs a zero byte. This ROM is usedto divide the coefficients of the error locator polynomial λ_(FINAL) (x)by the leading non-zero coefficient. ROM 1030 is a fast ROM. Inparticular, it receives a read address at its input 1048 and providesthe data from that address to its output 1050 in the same clock cycle.This timing synchronizes passage of coefficients through memory elements1016 and 1018 with the parallel passage of data through inverter path1056.

The coefficients λ_(FINAL) (x) are received at input 1044 from theoutput 530 of FIG. 5. The trigger signal applied to input 1010 is alsoused to inform the control circuit 1026 in which cycle the leadingcoefficient of λ_(FINAL) (x) is present at the input 1044. After beingsignalled that the coefficients of λ_(FINAL) (x) are being received, thecontrol circuit 1026 tests to determine when the leading non-zerocoefficient λ_(FINAL) (x) is received. The control circuit 1026 providesa first control signal to the multiplexor 1024 during the clock cycle inwhich the circuit 1026 detects on input 1044 the highest degree non-zerocoefficient of λ_(FINAL) (x). The second control signal to themultiplexor 1024 is provided during all other clock cycles. In responseto the first control signal, multiplexor 1024 passes the data from itsinput 1048 to its output. In response to the second control signal,multiplexor 1024 passes the data from its input 1050 to its output.Accordingly, multiplexor 1024 transmits the data from its input 1048 toits output only during the clock cycle in which the leading non-zerocoefficient of λ_(FINAL) (x) is present at its input 1048. In subsequentclock cycles, after this leading non-zero coefficient is stored inmemory element 1022, multiplexor 1024 couples the output of memoryelement 1022 to its input 1050 and the input 1050 to the input of memoryelement 1022 so that memory element 1022 maintains the leading non-zerocoefficient.

The output of the memory element (ROM)1030 is coupled to a first inputof a multiplexor 1034. The output of the memory element 1036 is coupledto the input 1054 of the multiplier 1020. The output of the multiplexor1034 is coupled to an input of the memory element 1036. The output ofthe memory element 1036 is also coupled by feedback path 1038 to thesecond input of multiplexor 1034. The control circuit 1032 provides thefirst control signal to the multiplexor 1034 during the clock cycle inwhich the reciprocal of the leading non-zero coefficient of the errorlocator polynomial becomes available at output 1050. Control circuit1032 provides a second control signal to the multiplexor 1034 after thisnon-zero coefficient has been stored memory element 1036 (i.e. in theclock cycle after the clock cycle in which the inverse becomes availableat output 1051). In response to the first control signal, multiplexor1034 passes the data from its first input to its output. In response tothe second control signal, multiplexor 1034 passes the data from itssecond input to its output. Accordingly, feedback path 1038 is operatingto maintain the inverse coefficient in memory element 1036 in the samemanner as was done by feedback path 1028. The output 1046 of themultiplier 1020 is the output of the standard error locator polynomialcalculator.

In operation, prior to being triggered by a trigger signal applied toinput 1010, control circuit 1040 configures multiplexor 1014 to couplethe input 1042 to the output 1046. Upon receiving the trigger signal atinput 1010, control circuit 1040 reconfigures multiplexor 1014 to coupleits input 1044 to its output 1046. When the first coefficient of errorlocator polynomial from calculator 310 is present at input 1044, it isapplied to the input of memory element 1016 and to input 1048. The firstclock cycle after this coefficient is provided at input 1044 thiscoefficient is clocked into memory element 1016. It will also be clockedinto memory element 1022 if it is non-zero. The control circuit 1026configures the multiplexor 1024 to couple the input 1050 to the output1058 until non-zero data is detected at input 1044. Next, the secondcoefficient of the error locator polynomial will be provided at input1044. Subsequent coefficients of the error locator polynomial areprovided at input 1044 with each clock cycle. Control circuit 1026 testseach of the coefficients on input 1044 to determine if it is non-zero.If it is zero, the control circuit 1026 continues to couple the input1050 to its output 1058. When the leading non-zero data is detected atinput 1044, input 1048 is coupled to output 1058 by control circuit 1026and the leading non-zero coefficient is latched into memory element1022. After this latching occurs, the control circuit 1026 againconfigures the multiplexor 1024 to couple its input 1050 to its output1058. Accordingly, memory element 1022 maintains storage of the firstnon-zero coefficient it receives even as subsequent coefficients arriveat input 1044. In particular, the data stored in memory element 1022will be unaffected by subsequent coefficients passed to input 1048 ofmultiplexor 1024 whether they are zero or non-zero.

Memory element 1030 receives this leading non-zero coefficient that isstored in memory element 1022 at its address input 1048. As we haveindicated, ROM 1030 provides at its output 1051 the reciprocal of thisnon-zero coefficient. This reciprocal output is latched into memoryelement 1036. After this latching, control circuit 1032 signalsmultiplexor 1034 to couple the output of memory element 1036 to itsinput using feedback path 1038. Accordingly, the reciprocal of theleading non-zero coefficient of the error locator polynomial ismaintained in memory element 1036. Memory element 1036 provides thisreciprocal coefficient to the input 1054 of multiplier 1020. Memoryelements 1016 and 1018 ensure that the coefficients of the error locatorpolynomial arriving at input 1052 of multiplier 1020 are properly timedwith the reciprocal data being latched into memory element 1036.Multiplier 1020 multiplies the error locator polynomial coefficientsreceived at the input 1052 by the reciprocal of the leading non-zerocoefficient of the error locator polynomial received at the input 1054.The coefficients of the standard error locator polynomial are outputfrom multiplier 1020 at output 1046.

Error Transform Calculator And Trigger Circuit

FIG. 11A illustrates the error transform calculator 314 used by thepresent embodiment. As discussed, the error transform calculator 314calculates the coefficients of the error transform polynomial E(x) fromthe coefficients of the syndrome polynomial S(x) and the coefficients ofthe standard error locator polynomial σ(x). It uses the relation##EQU10## where σ_(k) is the coefficient of the degree k term of thestandard error locator polynomial σ(x) and E_(16+j-k) is a knowncoefficient of the polynomial E(x). The coefficients E₀ through E₁₅(also called error transforms) of the polynomial E(x) are thecoefficients S₀ through S₁₅, respectively, of the syndrome polynomial.Accordingly, the error transform calculator starts with these knownerror transforms, E₀ through E₁₅, and calculates the remaining errortransforms E₁₆ to E₂₅₄.

This calculator includes eight error transform cells 1102 (i.e., 1102Ato 1102H). With respect to error transform calculators, the termspreceding cell and succeeding cell shall be defined according to thedirection of flow of the known error transforms through the cells. Eachcell includes a multiplexor 1104 (i.e. 1104A to 1104H), a memory element1122 (i.e. 1122A to 1122H), a standard error memory element 1134 (i.e.1134A to 1134H), a Galois field multiplier 1130 (i.e. 1130A to 1130H), amodulo two adder 1136 (i.e. 1136A to 1136H), a zero detector circuit1132 (i.e. 1132A to 1132H), and an AND gate 1128 (i.e.1128A to 1128H). Asyndrome input 1106 broadcasts the coefficients of the syndromepolynomial to all of the cells 1102. A logic zero input 1150 provideszero data to the first cell 1102A. A shift control line 1148 provides ashift control signal from the shift control circuit 1124 to all of thecells 1102. A logic one input 1126 provides logic one data to the firstcell 1102A. A logic zero input 1144 provides zero data to the last cell1102H. The output of the multiplexor 1154 provides the output 1156 ofthe error transform calculator 314. A control circuit 1146 controls themultiplexor 1154. A standard error memory element 1134I provides aninput to the last cell 1102H. All of the bus lines in the errortransform calculator are 8 bit lines except for the shift control line1148, the inputs and outputs of the AND gates 1128 and the control line1161. Line 1148, the inputs and outputs of the AND gates 1128 and lines1161 are one bit lines.

Within each cell, the multiplexor 1104 selects one of the three inputs1108, 1110, and 1112 to couple to its output 1118. The output 1118 ofmultiplexor 1104 is coupled to the input of memory element 1122. Memoryelements 1122 are 8 bit registers that provide to their outputs 1120 thedata stored in the memory element. When these memory elements areclocked by a clock signal (not shown), they store the data present attheir respective inputs. The input 1108 of multiplexor 1104 receivesdata from the syndrome input 1106. This input 1106 provides to each ofthe inputs 1108 the coefficients of the syndrome polynomial S(x). In thefirst cell 1102A, the input 1110A of multiplexor 1104 receives 8 bits ofzero data from the logic zero input 1150. In the cells 1102B through1102H, the input 1110 of the multiplexor 1104 is coupled to the output1120 of the memory element 1122 in the preceding cell 1102. Accordingly,the inputs 1110B to 1110H receive data from the outputs 1120A to 1120G,respectively. The output 1120H of the memory element 1122H in the lastcell 1102H is coupled to an input 1160 of the multiplexor 1154. Theinput 1112 of multiplexor 1104 receives data from the output 1142 of theadder 1136 in the same cell. For example, the input 1112A receives datafrom the output 1142A of adder 1136A.

The output 1120 of memory element 1122 is also coupled to a first inputof the multiplier 1130 in the same cell. For example, the output 1120Ais coupled to an input of the multiplier 1130A. The output of themultiplier 1130 is coupled to an input of the adder 1136 in the samecell. As we have noted the output of adder 1136 is coupled to the input1112 of the multiplexor 1104 in the same cell.

Control inputs 1114 and 1116 of multiplexor 1104 control which of theinputs 1108, 1110 and 1112 is coupled to the output 1118. Embodiments ofthe invention are not limited to multiplexors with two control inputs.Other embodiments other control input configurations to control theoperation of these multiplexors. Control inputs 1114 and 1116 are eachadapted to respond to first and second control signals. In the presentembodiment, the first control signal at input 1114 is a logic one leveland the second control signal is a logic zero level. Control input 1116is the same. The AND gate path 1158 could be modified to use othercontrol signals.

In the present embodiment, when both of the control inputs 1114 and 1116are at a logic one level, the multiplexor couples its input 1108 to itsoutput 1118. When the control input 1114 is at a logic zero level andthe control input 1116 is at a logic one level, the multiplexor 1104couples its input 1112 to its output 1118. When the input 1116 is at alogic zero level, the multiplexor 1104 couples its input 1110 to itsoutput 1118 regardless of the level on the 1114 input. The input 1114 ineach cell is coupled to the shift control line 1124. The input 1116 ineach cell is coupled to the AND gate path 1158. The AND gate path 1158includes AND gates 1128A to 128H coupled in series. Each AND gate 1128has two inputs and an output. A first input of each AND gate 1128 isprovided by the output of a zero detector circuit 1132 in the same cell.The zero detector circuit 1132 is coupled to standard error memoryelement 1134 in the same cell. The zero detector circuit 1132 provides asingle bit logic 1 output if the data in the same cell standard errormemory element 1134 is zero and a single bit logic 0 output if the datain the same cell standard error memory element 1134 is not zero. Thestandard error memory elements 1134 store the coefficients of thestandard error locator polynomial. In the first cell 1102A, the secondinput of the AND gate 128A is provided by the logic one input 1126. Incells 1102B to 1102H, the second input of the AND gate 1128 is providedby the output of the AND gate 1128 in the preceding cell. Thus, forexample, the second input of the AND gate 1128B is provided by theoutput of the AND gate 1128A. The AND gate 1128H and the zero detectorcircuit 1132H in the last cell 1102H are not used to drive a subsequentcell 1102. They have been included in the present embodiment, however,to maintain a "generic" cell design that is repeatedly reused,simplifying hardware design. This AND gate 1128H and the zero detectorcircuit 1132H might be removed in alternate embodiments. In the presentembodiment, the output of AND gate 1128 and the zero detector circuit1132H are used to properly configure the trigger circuit of FIG. 11B.

In the first cell, the control input 1116A of the multiplexor 1104A iscoupled to the logic one input 1126. In the cells 11102B to 1102H, theinputs 1116 are coupled to the output of the AND gate 1128 from thepreceding cell. Thus, the input 1116B is coupled to the output of theAND gate 1128A.

As a result of this configuration, the input 1116A always receives alogic one data bit. The logic level of the data bit received at theinputs 1116B to 1116H of the cells 1102B to 1102H, however, will dependon whether or not the data stored in the standard error memory elements1134A to 1134G, respectively, is zero as indicated by the output of thezero detector circuits 1132. In particular, starting with cell 1102A andmoving to each cell in the order in which they are coupled in series(e.g. 1102A to 1102B to 1102C . . . to 1102H), if the output of the zerodetector in the first cell is one, indicating that standard error memoryelement 1134 contains zero data, the output of the AND gate in that cellwill also be one. The output of AND gate 1128 will be one along theseries path until the standard error memory element 1134 in a cellcontains non-zero data.

When a standard error memory element 1134 contains non-zero data, theoutput of zero detector circuit 1132 in the same cell will be zero,making the output of the AND gate 1128 in the same cell zero. As aresult, the data received at the inputs 1116 in all of the succeedingcells 1102 will be zero regardless of the values stored in standarderror memory elements 1134 of these succeeding cells. In particular,zero data in the standard error memory elements 1134 in cells succeedingthe leading non-zero data cell will not cause an improper configurationof the multiplexors 1104. Any such zero coefficients shall be referredto as intermediate zero coefficients. Improper operation could result,for example, if an intermediate zero coefficient caused a high logiclevel to be applied to the control input 1116 of a cell succeeding thecell that contains the leading non-zero coefficient of σ(x). As theforegoing description illustrates, the control input 1116 of suchsucceeding cells should remain at a low logic level whether or not thestandard error memory element 1134 in these succeeding cells is zero.

The multiplexor 1104 in the cell that contains the non-zero data instandard error memory element 1134 and any preceding cells have a logicone applied to their control inputs 1116. This configuration by AND gatepath 1158 means that the multiplexors in all of the cells succeeding thefirst non-zero data cell will couple their input 1110 to their output1118. In particular, they will pass to their 1118 output data from apreceding cell that is provided at their 1110 input. The multiplexors1104 in the cell that contains the non-zero data and any preceding cellswill couple either their input 1108 or 1112 to their output 1118depending on the control signal present at input 1114. The multiplexor1104 in the cell that contains the non-zero data will not receive datafrom the preceding cells on the input 1110.

The standard error memory elements 1134B to 1134H are coupled back tothe preceding cell using lines 1138B to 1138H to provide a second inputto the multiplier 1130 of the preceding cell. The standard error memoryelement 1134B, for example, provides a second input to the multiplier1130A. An additional memory element 1134I is coupled using line 1138I tothe last cell 1102H to provide a second input to the multiplier 1130H inthat cell.

The control circuit 1146 controls whether the multiplexor 1154 providesdata from the logic zero input 1152 to the output 1156 or data from theinput 1160 to the output 1156. The control circuit provides a firstcontrol signal and a second control signal to the multiplexor. Inresponse to the first control signal, the multiplexor provides the datafrom the input 1160 to its output 1156. In response to the secondcontrol signal, the multiplexor provides the data from the logic zeroinput 1152 to the output 1156. The control circuit provides the firstcontrol signal to the multiplexor 1154 when the first error transformcoefficient becomes available at the input 1160 of the multiplexor 1154.The control circuit provides the second control signal to themultiplexor in the clock cycle following the one in which the last errortransform coefficient E₂₅₄ is available in memory element 1122H.

The error transform calculator 314 can be viewed as a plurality of errortransform processing circuits coupled in series. In the presentembodiment, a single error transform processing circuit includes themultiplexor 1104, memory element 1122, multiplier 1130, adder 1136 andinformation inputs 1108, 1110 and 1112 of a particular cell 1102. In thepresent embodiment, a single error transform processing circuit alsoincludes a standard error memory element 1134 of a cell immediatelysucceeding the particular cell. In other words, a particular errortransform processing circuit need not be completely within one cell.Embodiments of the invention are not limited to the particular errortransform processing circuits in the present embodiment. The controloperations of the error transform calculator 314 can be viewedcollectively as a control circuit. In the present embodiment, thiscontrol circuit includes the AND gates 1128, the zero detector circuits1132, the standard error memory elements 1134 and the shift control line1148. In the present embodiment, the standard error memory elements 1134are included in both the control circuit and the error transformprocessing circuits. Alternate embodiments need not have thisconfiguration. For example, alternate embodiments could use a first setof standard error memory elements in the error transform processingcircuit and a different set of standard error memory elements in thecontrol circuit. Alternate embodiments of the invention need not havethe same relationship between cells and components as the presentembodiment. In particular, alternate embodiments need not havecomponents grouped in a particular cell in the same manner as thepresent embodiment. The zero detector circuits 1132 and the standarderror memory elements 1134 can be viewed collectively as a zerodetector. Alternate embodiments of the invention can use other zerodetectors to accomplish the functions of the disclosed zero detector.

In operation, the coefficients of the standard error locator polynomialσ(x) are loaded into the memory elements 1134. The inputs to the memoryelements 1134 are not shown for convenience, but these coefficients arereceived from the output 1046 of the standard error locator polynomialcalculator 312. The memory element 1134A to 1134I store the coefficientsin order of descending degree. Accordingly, the memory element 1134Astores the highest degree coefficient, and the memory element 1134Istores the lowest degree coefficient. Thus, the AND gate path 1158 incombination with the zero detectors 1132 determine which memory element1134 contains the leading (highest degree) non-zero coefficient of thestandard error locator polynomial σ(x). This path 1158 configures themultiplexors 1104 of the cells 1102 in the manner described. Thus, themultiplexor in the leading non-zero coefficient cell 1102 will receiveits input either from the syndrome input 1106 or from the feedback pathprovided by the output 1142 of the adder 1136 in that same cell. Theshift line 1148 will control which input is received. The multiplexors1104 in the succeeding cells 1102 will receive their input from theoutput 1120 of the memory element 1122 in the preceding cell 1102.

As an example, assume that the coefficient σ₀ stored in memory element1134A is nonzero. A non-zero coefficient of σ(x) means that errors havebeen detected. As shown in FIG. 3, the delay 320 prevents the syndromesS₀ to S₁₅ from reaching the error transform calculator 314 until thecoefficients of the standard error locator polynomial σ(x) have beenloaded into the memory elements 1134. After these coefficients of σ(x)are loaded and the multiplexors are configured by the zero detectors1132 and the AND gate path 1158, the first syndrome S₀ reaches the input1106. At this time, the control circuit 1124 provides a high logic levelto control inputs 1114 of the multiplexors 1104. Because the input 1116Ais coupled to a high logic level, the multiplexor 1104A couples theinput 1108A to the output 1118A. Thus, in the cell 1102A, the firstsyndrome S₀ is applied to the input of the memory element 1122A throughthe multiplexor 1104A. This first syndrome is loaded into the firstmemory element 1122A when that memory element is clocked by a clockcircuit (not shown). In the succeeding cells 1102B to 1102H, however,multiplexors 1104B to 1104H couple their respective inputs 1110B to1110H to their respective outputs 1118B to 1118H. Accordingly, thememory elements 1122B to 1122H receive data from the memory elements1122 in the respective preceding cells 1102A to 1102G through therespective multiplexors 1104B to 1104H. In other words, the syndrome S₀applied to their inputs 1108 is ignored.

The memory element 1122A provides the error transform E₀ to its outputautomatically after E₀ is clocked into the memory element 1122A. Again,syndromes S₀ through S₁₅ are error transforms E₀ through E₁₅,respectively. Accordingly, prior to the second clock cycle, E₀ will beapplied to the input of the memory element 1122B in the cell 1102B fromthe output of memory element 1122A, and the second error transform E₁ isapplied to the input of memory element 1122A. The second clock cycleclocks the E₀ into the memory element 1122B and E₁ into the memoryelement 1122A. This sequence of operation continues until errortransform E₁₅ is clocked into memory element 1122A. During thissequence, the data received at the inputs 1112 of each multiplexor 1104is also ignored. When E₁₅ is clocked into memory element 1122A, memoryelement 1122H contains E₈.

After E₁₅ is clocked into memory element 1122A, the shift controlcircuit 1124 pulls the inputs 1114 to a logic low level. Control circuit1124 knows how many error transforms are being clocked in. Because thecells 1102B to 1102H which succeed the leading non-zero coefficient cell1102A have a low logic level at their inputs 1116, their operation isunchanged by this transition. In particular, they continue to receivedata from the preceding cell 1102 through their respective inputs 1110.The multiplexor 1104A in the leading non-zero coefficient cell 1102A,however, now couples the feedback input 1112A to the output 1118A. Thus,the data from the feedback input 1112A is now applied to the input ofthe memory element 1122A. As a result of the calculations performed bythe multipliers 1130 and the adders 1136, this data is the errortransform E₁₆. The next clock cycle will clock E₁₆ into the memoryelement 1122A. Memory elements 1122B to 1122H will now contain errortransforms E₁₅ to E₉, respectively. The data applied to the input 1112Aresulting from the calculations by multipliers 1130 and adders 1136 willbe E₁₇. The next clock cycle will clock E₁₇ into memory element 1122Aand E₁₆ into memory element 1122B, generating E₁₈ at the input 1112A.This process continues until E₂₅₄ is clocked into memory element 1122Hof the last cell 1102H.

Each data byte that is stored in memory element 1122H is provided to theoutput 1120H and is passed by multiplexor 1154 to output 1156. Each ofthese bytes is provided by the output 1156 to the input 1202 of theinverse error transform calculator 316 of FIG. 12. After E₂₅₄ isprovided out of output 1156, control circuit 1146 configures multiplexor1154 to pass zero bytes out of output 1156. Because in the presentexample the multiplexor 1104A does not accept data from the input 1112Auntil the error transform E₈ is latched into memory element 1122H, errortransforms E₀ through E₇ are passed to the inverse error transformcalculator 316 (FIG. 12), but are not used to calculate the unknownerror transforms E₁₆ through E₂₅₄ in the calculator 314 (FIG. 11A).Trigger circuit 1100 of FIG. 11B, discussed below, signals the inverseerror transform calculator 316 when the first error transform E₀ ispresent at the input 1202.

To demonstrate how the error transform calculator configures itselfdifferently if the leading non-zero coefficient of the standard errorlocator polynomial σ(x) is not in the memory element 1134A, assume thatthe leading non-zero coefficient of σ(x) is in the memory element 1134C.In other words, σ₀ and σ₁ are zero. Again, the coefficients of σ(x) areloaded into the memory elements 1134 and the multiplexors 1104 areconfigured by the zero detectors 1132 and the AND gate path 1158 beforethe first error transform E₀ reaches the input 1106. Like the previousexample, the control circuit 1124 provides a high logic level to controlinputs 1114 of the multiplexors 1104. Also like the previous example,the input 1116A is coupled to a high logic level, causing themultiplexor 1104A to couple the input 1108A to the output 1118A. Unlikethe previous example, however, the outputs of the zero detectors 1132Aand 1132B are both one because memory elements 1134A and 1134B,respectively, contain zero data. Accordingly, a logic one is applied atthe control inputs 1116B and 1116C. Thus, multiplexors 1104B and 1104Care also configured to couple their inputs 1108B and 1108C to theiroutputs 1118B and 1118C, respectively. Because the memory element 1134Ccontains the leading non-zero coefficient of σ(x), the output of zerodetector 1132C is zero. As a result, the inputs 1116D to 1116H allreceive a zero logic level, configuring multiplexors 1104D to 1104H tocouple their input 1110 to their output 1118.

When the error transform E₀ is received at the input 1106, it will beapplied to the inputs of the memory elements 1122A, 1122B and 1122Cthrough the multiplexors 1104A, 1104B and 1104C, respectively. E₀ isloaded into each of the memory elements 1122A, 1122B and 1122C whenthese memory elements are clocked by a clock circuit (not shown). In thesucceeding cells 1102D to 1102H, however, multiplexors 1104D to 1104Hcouple their respective inputs 1110D to 1110H to their respectiveoutputs 1118D to 1118H. Accordingly, the memory elements 1122D to 1122Hreceive data from the memory elements 1122 in the respective precedingcells 1102C to 1102G through the respective multiplexors 1104D to 1104H.Accordingly, the error transform E₀ applied to the inputs 1108D to 1108His ignored. The memory element 1122C provides the error transform E₀ toits output 1120C automatically after E₀ is clocked into the memoryelement 1122C. Accordingly, prior to the second clock cycle, thetransform E₀ will be applied to the input of the memory element 1122D inthe cell 1102D. Memory elements 1122A and 1122B also provide E₀ to theirrespective outputs 1120A and 1120B, but these outputs are ignored bymultiplexors 1104B and 1104C, respectively, which are configured toreceive their input from inputs 1108B and 1108C, respectively. Beforethe second clock cycle, the second error transform E₁ is applied to theinput of memory elements 1122A, 1122B and 1122C. The second clock cycleclocks the transform E₀ into the memory element 1122D and transform E₁into the memory elements 1122A, 1122B and 1122C. Again, the transform E₁is output from memory element 1122C and applied to the input of memoryelement 1122D while the outputs from memory elements 1122A and 1122B areignored. This sequence of operation continues until error transform E₁₅is clocked into memory elements 1122A, 1122B and 1122C. During thissequence, the data received at the inputs 1112 of each multiplexor 1104is also ignored. Unlike the previous example, when E₁₅ is clocked intomemory elements 1122A, 1122B and 1122C memory element 1122H contains E₁₀because the shift path along memory elements 1122 is shorter than in thefirst example. As described below, the trigger circuit 1100 of FIG. 11Badapts its delay to be shorter in correspondence to the length of theshift path in the calculator 314. Accordingly, even with the shortershift path, inverse error transform calculator 316 (FIG. 12) issignalled when the error transform E₀ is present at its input 1202.

After E₁₅ is clocked into memory elements 1122A, 1122B and 1122C, theshift control circuit pulls the inputs 1114 to a low logic level.Because the cells 1102D to 1102H which succeed the leading non-zerocoefficient cell 1102C have a low logic level at their inputs 1116,their operation is unchanged by this transition. They continue toreceive data from the preceding cell 1102 through their respectiveinputs 1110. The multiplexors 1104A, 1104B and 1104C, however, nowcouple their feedback inputs 1112A, 1112B and 1112C, respectively, totheir outputs 1118A, 1118B and 1118C, respectively. Thus, the data fromthe feedback inputs 1112A, 1112B and 1112C are now applied to the inputof the memory elements 1122A, 1122B and 1122C, respectively. As a resultof the calculations performed by the multipliers 1130 and the adders1136, the data at the input 1112C is the error transform E₁₆. The nextclock cycle will clock E₁₆ into the memory element 1122C. At this point,the data applied to the input 1112C will be E₁₇. Accordingly, the nextclock cycle will clock E₁₇ into memory element 1122C and E₁₆ into memoryelement 1122D. This process continues until E₂₅₄ is clocked into memoryelement 1122H of the last cell 1102H. As this cycle proceeds, the dataat the inputs 1112A and 1112B is unknown. It will be clocked into memoryelements 1122A and 1122B, respectively, but it will not affect thecalculation because multiplexor 1104C ignores data at its input 1110C.

Similar to the previous example, each data byte that is stored in memoryelement 1122H is provided to the output 1120H and is passed bymultiplexor 1154 to output 1156. Each of these bytes is provided by theoutput 1156 to the input 1202 of the inverse error transform calculator316 of FIG. 12. Because in the present example the multiplexor 1104Cdoes not accept data from the input 1112C until the error transform E₁₀is latched into memory element 1122H, error transforms E₀ through E₉ arepassed to the inverse error transform calculator 316 (FIG. 12), but arenot used to calculate the unknown error transforms E₁₆ through E₂₅₄. Theoperation of error transform calculator 314 changes in the same mannerwhen different memory elements 1134 contain the leading non-zerocoefficient of the standard error locator polynomial.

FIG. 11B is a variable delay trigger circuit 1100 that is used inconjunction with the error transform calculator 314. In particular, thistrigger circuit 1100 provides a variable delay that corresponds to thedelay associated with the configuration of the error transformcalculator 314 by the AND gate path 1158. This variable delay triggercircuit 1100 is used to pass the trigger signal output by the standarderror locator polynomial calculator 312 from output 1012 (FIG. 10) tothe control circuit 1220 of the inverse error transform calculator 316(FIG. 12). The delay by the trigger circuit 1100 adapts such that theinverse error transform calculator 316 (FIG. 12) is signalled when theerror transform E₀ is present at the input 1202.

The trigger circuit 1100 includes cells 1103 (i.e. 1103A-1103H). In thepresent embodiment, all of these cells include the same hardwareelements. In particular, each cell 1103 includes a memory register 1107(i.e. 1107A-1107H) and a multiplexor 1109 (i.e. 1109A-1109H). Thecontrol input 1115 (i.e. 1115A-1115H) of each multiplexor 1109 iscoupled to the AND gate path 1158 of FIG. 11A through the outputs 1119of the respective AND gates (as shown in FIG. 11B). These couplings arerepresented by the connectors 1111 (i.e. 1111A-1111H). Connectors 1111refer to the location of the connection into the AND gate path of FIG.11A. In particular looking at FIGS. 11A and 11B, each control input 1115is coupled to the output 1119 (i.e. 1119A-1119H) of the AND gate 1128 inthe corresponding cell. Thus, the output 1119A from the cell 1102A iscoupled to the control input 1115A in the cell 1103A; the output 1119Bfrom the cell 1102B is coupled to the control input 1115B in the cell1103B; and so on. Thus, when the memory element 1134A in cell 1102A ofcalculator 314 (FIG. 11A) contains a zero coefficient, for example, thememory element 11207A in the trigger circuit 1100 (FIG. 11B) is bypassedbecause the shift path in calculator 314 (FIG. 11A) through which thesyndromes are shifted from input 1106 to output 1156 will not includememory element 1122A. Under these circumstances, the output of AND gate1128A (FIG. 11A) provides a first control signal (high logic level inthe present embodiment) to the input of the multiplexor 1109A (FIG. 11B)so that the timing provided by the trigger circuit 1100 corresponds tothe timing of the calculator 314. Similarly, when the memory elements1134A and 1134B in cell 1102B of calculator 314 (FIG. 11A) contain azero coefficient, the memory elements 1107A and 1107B in the triggercircuit 1100 (FIG. 11B) are bypassed because the shift path incalculator 314 (FIG. 11A) from input 1106 to output 1156 through whichthe syndromes are shifted will not include memory elements 1122A and1122B. Under these circumstances, the output of the AND gates 1128A and1128B provide a first control signal (high logic level in the presentembodiment) to the control inputs of the multiplexors 1109A and 1109B.

If the memory element 1134A in calculator 314 contains a non-zerocoefficient, however, the memory element 1122A (FIG. 11A) will beincluded in the shift path of the calculator 314 through which thesyndromes are shifted. Accordingly, memory element 1107A (FIG. 11B)should be included in the trigger circuit 1100 shift path so that thetiming of the trigger circuit 1100 correctly corresponds to the timingof the error transform calculator 314. Under these circumstances, theAND gate 1128A will output a second control signal (low logic level inthe present embodiment) that will cause the multiplexor 1109A to passthe data from the memory element 1107A. This second control signalresults because the output of the zero detector 1132A will be a lowlogic level when non-zero data is present in memory element 1134A.Because the output of AND gate 1128A is a low logic level, the output ofall succeeding AND gates 1128 will also be a low logic level.Accordingly, all succeeding trigger circuit cells 1103 will pass datafrom the memory elements 1107. Thus, all succeeding memory elements 1107will contribute to the delay of the trigger circuit 1100. Such a delaycorresponds to the delay resulting from the error transform calculator314 as configured by the AND gate path 1158.

If memory element 1134A had contained a zero coefficient and memoryelement 1134B had contained the first non-zero coefficient of thestandard error locator polynomial, memory elements 1107B through 1107Hbut not 1107A would be included in the trigger circuit 1100 shift pathso that the delay caused by trigger circuit 1100 would correspond to thedelay of the shift path in calculator 314 (FIG. 11A) from input 1106 tooutput 1156. Similarly, if memory element 1134C of the calculator 314(FIG. 11A) contained the first non-zero coefficient, then memoryelements 1107C through 1107H but not 1107A and 110B would be included inthe shift path of the trigger circuit 1100. Thus, in general triggercircuit 1100 includes the memory elements 1107 in its shift pathstarting with the cell 1103 associated with the cell 1102 that containsthe first non-zero coefficient of the standard error locator polynomialand including the memory elements 1107 of any cells 1103 succeeding thefirst non-zero coefficient cell. Trigger circuit 1100 does not includein its shift path the memory elements 1107 in any cells 1103 precedingthe first non-zero coefficient cell. Alternate embodiments of theinvention may use varying numbers of cells to form the appropriate shiftpaths. Feedback loops could also be used.

Inverse Error Transform Calculator

FIG. 12 illustrates the inverse error transform calculator 316 used bythe present embodiment. While N-1=254 in the present embodiment, theinverse error transform polynomial need only calculate inverse errortransforms e₀ to e₂₀₃ because of the use of the shortened (204, 188) RScode. The present embodiment need not calculate the transforms e₂₀₄ toe₂₅₄ because they relate to the unused data bytes described earlier. Thepresent embodiment does not need to detect and correct errors in theseunused data bytes. Alternate embodiments could be designed to detect andcorrect errors in these additional data bytes if desired.

Because the present embodiment need only calculate e₀ to e₂₀₃, memoryelements 1206 in the inverse error transform calculator 316 store onlyα.sup. for 0≦n≦203. Inverse error transform calculator 316 calculatesinverse error transforms using the equation ##EQU11## where E_(k) is thecoefficient of the degree k term of the error transform polynomial E(x).In the present embodiment, ##EQU12## even though N-1=254 because of theuse of the shortened Reed Solomon code.

The inverse error transform calculator 316 includes 204 cells 12040 to1204₂₀₃. Each cell 1204 includes a memory element 1206 (i.e. memoryelements 1206₀ to 1206₂₀₃), a memory element 1208 (i.e. 1208₀ to1208₂₀₃), a memory element 1216 (i.e. 1216₀ to 1216₂₀₃), a modulo twoadder 1210 (i.e. 1210₀ to 1210₂₀₃), a Galois field multiplier 1212 (i.e.1212₀ to 1212₂₀₃) and a multiplexor 1214 (i.e. 1214₀ to 1214₂₀₃). Thecalculator 316 also includes an error transform input 1202, an inverseerror transform output 1226 and a control line 1232. Memory elements1206, 1208 and 1216 are 8 bit registers that provide the data that theyare storing to their outputs. When these memory elements are clocked bya clock signal (not shown), they store the data present at theirrespective inputs. All of the bus lines in calculator 316 are 8 bit widebus lines, except the control line 1232, which is a one bit line. Thecontrol circuit 1220 provides control signals to the inverse errortransform calculator 316 using control line 1232.

The control circuit 1220 provides a first control signal and a secondcontrol signal on control line 1232. The control line 1232 is coupled toa control input 1234 of each multiplexor 1214. In response to the firstcontrol signal, the multiplexors 1214 couple their input 1238 to theiroutput 1240. In response to the second control signal, the multiplexors1214 couple their input 1236 to their output 1240. In the presentembodiment, the first control signal is a logic 1 and the second controlsignal is a logic 0.

The error transform input 1202 is coupled to the input of all of thememory elements 1208. This input 1202 provides the error transformscalculated by the error transform calculator 314 to the memory elements1208. The outputs of the memory elements 1208 are coupled to a firstinput of the modulo two adder 1210 in the same cell. Memory elements1206 store decreasing powers of α starting with α²⁰³ in cell 1204₂₀₃ andending with α⁰ or 1 in cell 1204₀. Memory elements 1206 are 8 bitregisters. They may be of the type that accepts clocked data from aninput, or some kind of ROM, for example. The outputs of memory elements1206 are coupled to a first input of the multiplier 1212 in the samecell. The second input to the multiplier 1212 is coupled to the output1228 of memory element 1216 in the same cell. The output of themultiplier 1212 is coupled to the second input of the adder 1210 in thesame cell. The output of the adder 1210 is coupled to the input 1236 ofthe multiplexor 1214 in the same cell. The output 1240 of themultiplexor 1214 is coupled to the input of the memory element 1216 inthe same cell. In each preceding cell, the output 1230 of the memoryelement 1216 is coupled to the input 1238 of the multiplexor 1214 in thesucceeding cell. In the last cell 1204₀, the output 1230₀ of memoryelement 1216₀ is coupled to a first input to the multiplexor 1224. Azero data input 1222 is coupled to an input of the multiplexor 1224. Azero data input 1218 is coupled to the input 1238₂₀₃ of multiplexor1214₂₀₃.

Prior to receiving the first error transform at input 1202, the memoryelement 1216 of each cell is initiallized to zero. During the normalcourse of operation, the memory elements 1216 are automaticallyinitiallized when the evaluated inverse error transform coefficients areshifted out of memory elements 1216. At this time, the control circuit1220 configures the multiplexors 1214 to couple their zero input 1218 totheir output 1240 so that zero values are shifted into memory elements1216 when the inverse error transform coefficients are shifted out.

After initialization, the control circuit 1220 applies the first controlsignal to line 1232, configuring the multiplexors 1214 to couple theirinputs 1236 to their respective outputs 1240 and configuring multiplexor1224 to couple the zero valued input 1222 to the output 1226. Output1156 of FIG. 11A is coupled to input 1202 of FIG. 12.

The error transforms from the output 1156 of the error transformcalculator 314 are received at the input after the calculator 316 isinitialized in this manner. The trigger circuit 1100 informs thecalculator 316 when the first error transform E₀ is provided at theinput 1202. At this point, a first clock cycle stores the first errortransform E₀ in the memory elements 1208. The sum of the data stored inmemory element 1208 and the output of multiplier 1212 is applied to theinput of the memory element 1216 from the output of adder 1210. Theoutput of multiplier 1212 is the product of the power of α stored inmemory element 1206 and the data stored in memory element 1216. Afterthis first clock cycle, the data in memory element 1216 is still zero.Accordingly, the output of adders 1210 is E₀. The second error transformis then applied to the input of memory element 1208.

The second clock cycle stores E₀ (i.e. the sum present at the input ofmemory element 1216) in memory element 1216 and stores the second errortransform E₁ in memory element 1208. The sum of the data stored inmemory element 1208 and the output of multiplier 1212 is again appliedto the input of the memory element 1216. This sum is E₁ +E₀ α^(n) wheren is the power of α stored in the memory element 1206 of the particularcell. The output of multiplier 1212 is the product of the power of αstored in memory element 1206 and the data stored in memory element1216. At this point, the data stored in memory element 1216 is E₀. Afterthe next clock cycle the sum E₂ +E₁ α^(n) +E₀ α²(n) is applied to theinput of memory element 1216_(n). The cells 1204 are generating multiplepowers of α. The calculator 316 is clocked 255 times so that all of E₀through E₂₅₄ are summed into memory elements 1216. On the 256th clock, adata zero value is stored in memory elements 1208 from the output 1156.At this point, the calculator 316 is clocked one additional time (i.e.the 257th time). Because memory elements 1208 contain a zero, the datavalues stored in memory elements 1216 are multiplied by their respectiveα^(n) stored in the respective memory elements 1206. After this finalclocking, memory element 1216₀ contains e₀, memory element 1216₁contains e₁, memory element 1216₂ contains e₂, memory element 1216₃contains e₃ and memory elements 1216₄ through 1216₂₀₃ contains e₄through e₂₀₃, respectively.

The present embodiment uses the relation ##EQU13## to calculate theinverse error transforms. The present embodiment has been able to usethis relation to eliminate a buffer that might otherwise have increasedthe chip area required by the decoder 202. In particular, the inverseerror transform polynomial e(x) can be calculated from the errortransform polynomial E(x) using the relation ##EQU14## Please see, forexample, Shao, H, et al., A VLSI Design of a Pipeline Reed-SolomonDecoder, IEEE Transactions on Computers, Vol. C-34, No. 5, May 1985 page401, equation (21). This equation (23) could be calculated using acalculator such as calculator 316 that relies upon a recursive approachto calculate α^(-nk) where 0≦n≦N-1. In particular, memory elements 1206could store the constants α^(-n) instead of the constants α^(n)described above. The "k" exponent could be calculated using recursivemultiplications, as is done in the calculator 316 as is also describedabove. Unfortunately, to calculate equation (23) in this manner usingcalculator 316 where memory elements 1206 store α^(-n), E₂₅₄ would haveto be provided to input 1202 first, followed in order by errortransforms E₂₅₃, E₂₅₂, and E₂₅₁ through E₀. This order would be requiredbecause equation (23) requires that E₂₅₄ be multiplied by α^(-254n). Inthe contemplated calculator the "254" portion of the exponent isgenerated using recursive multiplications. Accordingly, if E₀ enteredthe calculator input 1202 first, E₀ would be multiplied by α^(-254n)which would not provide the correct result.

As we have described with respect to FIG. 11, the error transforms areoutput from the error transform calculator 314 output 1156 in the orderE₀ through E₂₅₄. A buffer could be used to reverse the order of theerror transforms applied to the input 1202 (i.e. to apply E₂₅₄ first).Such a buffer would sacrifice speed and require extra space on a chip.In particular, such a buffer would likely require at least 255 cycles tostore the 255 error transforms and an additional 255 cycles to accessthem after storage. In the present embodiment, such a buffer wouldrequire 255 eight bit registers. This embodiment has combined the use ofthe derived equation (22) and the sequence in which error transforms areoutput from calculator 314 to eliminate the need for such a buffer. Theapproach used by the present embodiment adds only one additional clockcycle (i.e. the 257th clock cycle) to the minimum number of cyclesrequired to generate the inverse error.

The following illustrates the derivation of this equation. Inparticular, both sides of equation (23) are multiplied by (α^(n))^(N-1)to produce ##EQU15## Next, (α^(n))^(N-1) e_(n) =(α^(Nn-n))e_(n). Due tothe cyclic nature of Reed-Solomon codes, however, we know that(α^(N))=1. Thus, (α^(n))^(N-1) e_(n) =(α^(Nn-n))e_(n) =(1*α^(-n))e_(n).In other words, ##EQU16## Multiplying both sides by α^(n) produces##EQU17## which is the equation used by calculator 316 to calculatee_(n).

As calculator 316 demonstrates, it is calculating ##EQU18## again due tothe shortened RS code. In particular, using cell 203 as an example, E₀initially is clocked into memory element 1216₂₀₃ . In the next clockcycle, memory element 1216₂₀₃ will contain E₀ α²⁰³ +E₁. In the nextclock cycle, memory element 1216₂₀₃ will contain E₀ α²⁰³(2) +E₁ α²⁰³+E₂. This process continues until memory element 1216₂₀₃ is clocked tocontain E₀ α²⁰³(254) +E₁ α²⁰³(253) +E₂ α²⁰³(252) + . . . +E₂₅₄ α²⁰³(0).In a similar manner, the calculator that each of the memory elements1216₂₀₂ through 1216₀ will contain a respective summation ##EQU19## forthe values of n from 202 to 0, respectively. In the same clock cyclethat the final sum is stored in each of the memory elements 1216, zerosare stored in memory elements 1208. The calculator 316 is then clockedone additional time to multiply by α^(n) the series ##EQU20## as storedin memory elements 1216. As a result, the memory elements 1216 willcontain e₂₀₃ in cell 1204₂₀₃, e₂₀₂ in cell 1204₂₀₂, and so on down to e₀in cell 1204₀ as shown in equation (24).

After the inverse error transforms e₀ through e₂₀₃ are calculated, theyare clocked out of output 1226 and applied to a first input of themodulo two adder 306 illustrated in FIG. 3. The second input to themodulo two adder is the received data stream r(x)=r₀ x²⁰³ +r₁ x²⁰² + . .. +r₂₀₃ x⁰. The modulo two adder adds e₂₀₃ to r₀ to produce t₀, e₂₀₂ tor₁ to produce t₁ and e₂₀₁ through e₀ to r₂ through r₂₀₃ to produce t₂through t₂₀₃ Again, t₀ through t₂₀₃ provide the transmitted data stream.This transmitted data stream is then provided to an input of converter318 where it is converted in a manner known in the art to thepre-encoded data stream I(x).

Galois Field Multiplier

The present embodiment of the invention uses the Galois field multiplier1300 shown in FIG. 13 to accomplish Galois field multiplications. Thismultiplier 1300 enables computation of a Galois field product in lessthan one clock cycle. Other Galois field multipliers could be used inembodiments of the invention by making any appropriate timingadjustments to the decoder circuits. The Galois field multiplier 1300can be used to perform Galois field multiplication in applications otherthan the decoder circuits disclosed herein.

In general, embodiments of this multiplier can be used to multiplywithin a Galois field a first m bit word with a second m bit word. Thenumber m is the degree of a field generator polynomial of the Galoisfield GF(2^(m)). In the present embodiment m is 8. Multiplier 1300 usesa normal basis and a dual basis of the Galois field GF(2⁸) to accomplishthe Galois field multiplication. Galois fields have a normal basis{α^(k) } for 0≦k≦m-1 and a dual basis {λ_(k) }for 0≦k≦m-1. Again, α is aroot of the field generator polynomial. For a discussion of themathematics used by Galois field multipliers, please see Hsu et al, TheVLSI Implementation of a Reed-Solomon Encoder Using Berlekamp'sBit-Serial Multiplier Algorithm, IEEE Transactions on Computers, Vol.C-33, No. 10, October 1984 and Berlekamp, Bit-Serial Reed-SolomonEncoders, IEEE Transactions on Information Theory, Vol. IT-28, No. 6,November 1982 both of which are hereby incorporated herein by thisreference.

Multiplier 1300 includes inputs 1302 and 1304, a normal to dual basisconverter circuit 1306, a multiplier circuit 1308 having an output 1310and a dual to normal basis converter circuit 1312 having an output 1314.The present multiplier 1300 is designed to work with 8 bit words. Thebus lines in FIG. 13 are 8 bit bus lines. Alternate embodiments,however, are not limited to any particular word size.

In FIG. 13, an m bit word G is input into the input 1302, and an m bitword Z is input into the input 1304. Each of these words G and Z can berepresented using the normal basis {α^(k) } for 0≦k≦m-1 and the dualbasis {λ_(k) } for 0≦k≦m-1 of the Galois field GF(2^(m)). For example,the word G has the normal basis representation G=Gbit_(m-1) α^(m-1)+Gbit_(m-2) α^(m-2) + . . . +Gbit₀ α⁰ in the Galois field GF(2^(m)). Theshorthand notation {G_(m-1) . . . G₀ } shall be used to refer to thisnormal basis representation. In this shorthand, G_(m-1) =Gbit_(m-1)α^(m-1), G_(m-2) =Gbit_(m-2) α^(m-2), . . . G₀ =Gbit₀ α⁰ where Gbit_(x)refers to the bit in the x position of the word G and where bit zero isthe bit in the least significant bit position. Thus, where m=8, the wordG might be the byte Gbit₇ Gbit₆ Gbit₅ Gbit₄ Gbit₃ Gbit₂ Gbit₁ Gbit₀, forexample. This word G can be expressed using the normal basisrepresentation as G=Gbit₇ α⁷ +Gbit₆ α⁶ + . . . +Gbit₀ α⁰. From thisnormal basis representation, it can be seen that G₇ =Gbit₇ α⁷, G₆ =Gbit₆α⁶, . . . G₀ =Gbit₀ α⁰.

The word G can also be represented in the dual basis. Again, when m=8,the word G is the byte Gbit₇ Gbit₆ Gbit₅ Gbit₄ Gbit₃ Gbit₂ Gbit₁ Gbit₀,for example. The word G has a corresponding word G_(d) that representsthe dual basis representation of the word G. When m=8, the word G_(d) isthe byte G_(d) bit₇ G_(d) bit₆ G_(d) bit₅ G_(d) bit₄ G_(d) bit₃ G_(d)bit₂ G_(d) bit₁ G_(d) bit₀, for example. Using this word G_(d), the wordG can be expressed using the dual basis representation G=G_(d) bit_(m-1)λ_(m-1) +G_(d) bit_(m-2) λ_(m-2) + . . . +G_(d) bit₀. The shorthandnotation {G_(dm-1) . . . G_(d0) } shall be used to refer to this dualbasis representation. From this representation, it can be seen thatG_(dm-1) =G_(d) bit_(m-1) λ_(m-1) G_(dm-2) =G_(d) bit_(m-2) λ_(m-2) . .. G_(d0) =G_(d) bit₀ λ₀ where G_(d) bit_(x) refers to the bit in the xposition of the word G_(d) and where bit zero is the bit in the leastsignificant bit position. Thus, G_(d7) =G_(d) bit₇ λ₇, G_(d6) =G_(bit) ₆λ₆ and so forth. The λ values used in the dual basis polynomial areelements of the dual basis. These λ values should not be confused withthe earlier λ elements used in the context of the error locatorpolynomial and the λ-μ calculator. The present λ values are unrelated tothe earlier λ elements. In the present embodiment, the dual basis thatis used is the dual basis related to the field generator polynomialp(x)=x⁸ +x⁴ +x³ +x² +1 from the ETS 300 429 specification.

In the multiplier 1300, the m bit word G is input directly to themultiplier circuit 1308 using the input 1302. The normal to dual basisconverter circuit 1306 receives any m bit word that corresponds to anormal basis representation of the any m bit word (e.g. the word Zcorresponds to the normal basis representation of Z which is Zbit_(m-1)α^(m-1) +Zbit_(m-2) α^(m-2) + . . . +Zbit₀ α⁰). It converts the any mbit normal basis word into an m bit dual basis word (e.g. it convertsthe normal basis word Z into the dual basis word Z_(d) where Z_(d)corresponds to the dual basis representation of Z which is Z_(d)bit_(m-1) λ_(m-1) +Z_(d) bit_(m-2) λ_(m-2) + . . . +Z _(d) bit₀ λ₀).Thus, in the present embodiment, the m bit word Z that is input into theinput 1304 is converted by the normal to dual basis converter circuit1306 to the dual basis word Z_(d) which corresponds to the dual basisrepresentation {Z_(d0) . . . Z_(dm-1) } of the word Z. When m=8, theword Z_(d) is the byte Z_(d) bit₇ Z_(d) bit₆ Z_(d) bit₅ Z_(d) bit₄ Z_(d)bit₃ Z_(d) bit₂ Z_(d) bit₁ Z_(d) bit₀, for example. Again, theexpression {Z_(dm-1) . . . Z_(d0) } is a shorthand representation of thepolynomial

    Z=Z.sub.d bit.sub.m-1 λ.sub.m-1 +Z.sub.d bit.sub.m-2 λ.sub.m-2 + . . . +Z.sub.d bit.sub.0 λ.sub.0(25)

where Z_(d) bit_(x) refers to the bit in the x position of the dualbasis word Z_(d). In this dual basis representation, Z_(d0) =Z_(d) bit₀λ₀, Z_(d1) =Z_(d) bit₁ λ₁, . . . Z_(dm-1) =Zbit_(m-1) λ_(m-1). Afterconversion, the dual basis word Z_(d) is input into the multipliercircuit 1308. The elements of the dual basis representation {Z_(d0) . .. Z_(dm-1) } shall be referred to generically and/or collectively asZ_(dk).

The multiplier circuit 1308 is adapted to generate from the normal basisword G and from the dual basis word Z_(d) a dual basis product GZ_(d)which is the byte GZ_(d) bit₇ GZ_(d) bit₆ GZ_(d) bit₅ GZ_(d) bit₄ GZ_(d)bit₃ GZ_(d) bit₂ GZ_(d) bit₁ GZ_(d) bit₀. This dual basis product GZ_(d)can be generated by multiplying a binary mapping vector by each of m"dual basis vectors" where the m dual basis vectors have the form {Z_(d)bit_(0+p) . . . Z_(d) bit_(m-1+p) }. Each of the m "dual basis vectors"{Z_(d) bit_(0+p) . . . Z_(d) bit_(m-1+p) } has elements determined byone value of p as p varies across 0≦p≦m-1. In particular, the m "dualbasis vectors" are defined by the m values of p from 0 to m-1. Thus, inthe present embodiment a first dual basis vector (for p=0) has theelements {Z_(d) bit₀ . . . Z_(d) bit_(m-1) }. A second dual basis vector(for p=1) has the elements {Z_(d) bit₁ . . . Z_(d) bit_(m) }, forexample. The m^(th) dual basis vector (for p=m-1) has the elements{Z_(d) bit_(m-1) . . . Z_(d) bit_(2m-2) }. Again, the terms Z_(d) bit₀ .. . Z_(d) bit_(m-1) are bits of the word Z_(d). The terms Z_(d) bit_(m). . . Z_(d) bit_(2m-2) are not bits of the word Z_(d), however, but are"additional" bits determined from the bits Z_(d) bit₀ . . . Z_(d)bit_(m-1). In embodiments of the present invention, the values for atmost m-1 "additional vector" bits (e.g. Z_(d) bit_(m) . . . Z_(d)bit_(2m-2)) will be determined to define the m vectors. Where m=8, atmost seven additional bits will be determined to define 8 vectors. Theterm "dual basis vectors" is used for descriptive convenience. Itincludes any vectors that accomplish the function described herein. Eachmultiplication of the binary mapping vector by one of the dual basisvectors produces a bit of the dual basis product GZ_(d).

To determine the "additional vector" bits of the dual basis vectors, thesecond dual basis vector is generated from the first dual basis vectorby substituting the byte Y where Y=α Z_(d) into the first dual basisvector. Thus, Z_(d) bit₀ is replaced with Ybit₀, Z_(d) bit₁ is replacedwith Ybit₁ and so on. This substitution is shown in FIG. 16A. Inparticular, the first dual basis vector (vector 0) is numbered 1604₀.Substituting the byte Y into the first vector 1604₀ produces the vector1614₁. Using the relation 1606 (Y=α Z_(d)), the "Ybit" elements of thevector 1614₁ can be expressed in terms of the known bits of the wordZ_(d). In particular, the equations 1608 of FIG. 16A demonstrate thatYbit₀ equals Z_(d) bit₁, Ybit₁ equals Z_(d) bit₂ . . . and Ybit₆ equalsZ_(d) bit₇. As shown in equations 1608, Ybit₇ is equal to what has beenlabelled Z_(d) bit₈. Z_(d) bit₈ is equal to Tr(α⁸ Z_(d)). It can bedetermined using finite field arithmetic that α⁸ =α⁰ +α² +α³ +α⁴. Thisequivalence can be seen from the mapping shown in FIGS. 15A and 15Bwhich is discussed below. Thus, Z_(d) bit₈ can be expressed in terms ofthe bits of the word Z_(d) as Z_(d) bit₈ =Tr(α⁸ Z_(d))=Tr((α⁰ +α² +α³+α⁴)Z_(d))=Z_(d) bit₀ +Z_(d) bit₂ +Z_(d) bit₃ +Z_(d) bit₄. Accordingly,all of the elements of the second dual basis vector 1604₁ can beexpressed in terms of the bits of the word Z_(d). In the presentembodiment, the relation between the bits of the word Z_(d) and Z_(d)bit₈ has been identified prior to run time of the circuit and even priorto design of the circuit. This relation enables a fast circuit design.Alternate embodiments need not identify these mappings prior to runtime. The mappings between the bits of the word Z_(d) and each of thebits {Z_(d) bit_(m-1) . . . Z_(d) bit_(2m-2) } are individual mappingsthat in the present embodiment are implemented in parallel. Alternateembodiments may implement less than all of these mappings in parallelwith each other.

The third dual basis vector 1604₂ is determined from the first dualbasis vector 1604₀ in a similar manner. In particular, to generate thevector 1604₂ the byte Y is substituted into vector 1604₀ to produce thevector 1614₂. When generating the vector 1604₂, however, the relation1610 (y=α² Z_(d)) is used. The equations 1612 demonstrate the expressionof each of the Ybit elements of vector 1614₂ in terms of the bits of theword Z_(d) when using the relation 1610. In this case, Ybit₆ =Z_(d)bit₈. Because Z_(d) bit₈ was expressed in terms of the bits of the wordZ_(d) with respect to vector 1614₁, that relation need not be rederived.As shown in equations 1608, Ybit₇ is equal to what has been labelledZ_(d) bit₉. Thus, to express the elements of vector 1614₂ in terms ofthe bits of the word Z_(d), only one new relation needs to be evaluated.In particular, Z_(d) bit₉ is equal to Tr(α⁹ Z_(d)). It can be determinedusing finite field arithmetic that α⁹ =α¹ +α³ +α⁴ +α⁵. This equivalencecan be seen from the mapping shown in FIGS. 15A and 15B which isdiscussed below. Thus, similar to Z_(d) bit₈, Z_(d) bit₉ can beexpressed in terms of the bits of the word Z_(d) as Z_(d) bit₉ =Tr(α⁹Z_(d))=Tr((α¹ +α³ +α⁴ +α⁵)Z _(d))=Z_(d) bit₁ +Z_(d) bit₃ +Z_(d) bit₄+Z_(d) bit₅. Thus, all of the elements of the third dual basis vector1604₂ can be expressed in terms of the bits of the word Z_(d). Similarto the expression of Z_(d) bit₈, the relation between the bits of theword Z_(d) and Z_(d) bit₉ has been identified prior to run time of thecircuit and even prior to design of the circuit. This relation enables afast circuit design.

The remaining dual basis vectors are shown in FIG. 16B. These vectorsare generated in a similar manner to the vectors 1604₁ and 1604₂, exceptthat for each of the vectors 1604_(p), the relation Y=α^(p) Z_(d) isused. Thus, vector 1604₃ is generated using the relation Y=α³ Z_(d),vector 1604₄ is generated using the relation Y=α⁴ Z_(d) and so on.Because the elements Z_(d) bit₈ and Z_(d) bit₉ have been expressed interms of the bits of the word Z_(d) when generating vectors 1604₁ and1604₂, these elements need not be reevaluated. Thus, to express theremaining vectors 1604₃ to 1604₇ in terms of the bits of the word Z_(d),relations between the bits of the word Z_(d) and elements Z_(d) bit₁₀,Z_(d) bit₁₁, Z_(d) bit₁₂, Z_(d) bit₁₃ and Z_(d) bit₁₄ are evaluated.These relations (also called individual mappings) are generated in thesame manner described with reference to vectors 1604₁ and 1604₂. Theresults of these evaluations is shown in FIG. 16B.

In the present embodiment, the binary mapping vector has elements thatare the bits of the word G. In the present embodiment, the binarymapping vector is a "1 row by 8 column" vector that contains the leastsignificant bit of the word G in the first column. Subsequent columnscontain the increasingly significant bits of the word G up to the mostsignificant bit of the word G in the last column. Thus, in the presentembodiment, the binary mapping vector is a 1 by 8 vector having elements{Gbit₀ Gbit₁ Gbit₂ Gbit₃ Gbit₄ Gbit₅ Gbit₆ Gbit₇ }.

In the present embodiment, the multiplier circuit 1308 generates the mbits of the dual basis product GZ_(d) by multiplying the "1×m" binarymapping vector by each of the individual "m×1" dual basis vectors {Z_(d)bit_(0+p) . . . Z_(d) bit_(m-1+p) } for each of the values of p as pvaries across 0≦p≦m-1. These multiplications are illustrated in FIG.16C. Mod(2) arithmetic is used during the multiplication process.Multiplication of the word G by the dual basis vector {Z_(d) bit_(0+p) .. . Z_(d) bit_(m-1+p) }for p=0 generates the bit zero of the productGZ_(d) or GZ_(d) bit₀. Similarly, multiplication of the word G by thedual basis vector {Z_(d) bit_(0+p) . . . Z_(d) bit_(m-1+p) }for p=1generates GZ_(d) bit₁. Multiplication of the word G by the individualdual basis vectors for the remaining values of p up to m-1 generates theremaining bits of the product GZ_(d) or the bits GZ_(d) bit_(p). For afurther discussion of the mathematics underlying the binary mappingvector and the m dual basis vectors {Z_(d) bit_(0+p) . . . Z_(d)bit_(m-1+p) }, please see Hsu et al, The VLSI Implementation of aReed-Solomon Encoder Using Berlekamp's Bit-Serial Multiplier Algorithm,IEEE Transactions on Computers, Vol. C-33, No. 10, October 1984 andBerlekamp, Bit-Serial Reed-Solomon Encoders, IEEE Transactions onInformation Theory, Vol. IT-28, No. 6, November 1982. While thismultiplication is described in terms of vectors, the multiplication isintended to encompass alternate approaches that accomplish the sameresult. Thus, all of the dual basis vectors may be combined into a large8 by 8 dual basis matrix and the binary mapping vector is multiplied bythis matrix. In fact, such an approach still multiplies the identifiedvectors. Similarly, the multiplied bits may not be labelled or groupedas vectors. Even if handled on an individual basis, however, multiplyingthe individual bits as is done in the vector multiplication isconsidered as performing the multiplication of the identified vectors.

Once the multiplier circuit 1308 generates the dual basis productGZ_(d), the dual basis product is provided to a dual to normal basisconverter circuit 1312 using the output 1310. In the present embodiment,the dual to normal basis converter circuit 1312 converts any 8 bit wordthat corresponds to a dual basis representation into an 8 bit word thatcorresponds to the normal basis representation. The dual to normal basisconverter circuit 1312 can be adapted to handle m bit words fordifferent values of m. In the present embodiment, this circuit 1312 isused to convert the dual basis product GZ_(d) into the normal basisproduct GZ in the Galois field GF(2⁸).

The designs of the normal to dual basis converter circuit 1306, of themultiplier circuit 1308 and of the dual to normal basis convertercircuit 1312 enable the Galois field multiplier 1300 to produce thenormal basis product GZ in under one clock cycle from the time that themultiplier G and the multiplicand Z are applied to the inputs 1302 and1304, respectively. The multiplier 1300 could be adapted to provide thenormal basis product in less than any number of clock cycles (e.g. lessthan any one of 8, 7, 6, 5, 4, 3, 2, 1 clock cycles). Typically indevices doing Reed-Solomon calculations in an actual application,shorter delays are better. The multiplier 1300 advantageously enablesthe normal basis product GZ to be generated in less than the m clockcycles that might be required by a conventional serial multiplier.

FIGS. 14A-14F are functional diagrams of an embodiment of the normal todual basis converter circuit 1306. These figures illustrate that theinput from input 1304 and the output to multiplier circuit 1308 are 8bit buses. The separate FIGS. 14A-14F have been used for ease ofillustration to illustrate the mappings from the input pins to theoutput pins of the converter circuit 1306. The bits of the word Z areinput to the inputs 1402. The bits of the word Z_(d) are output from theoutputs 1404.

The normal to dual basis converter circuit 1306 implements a generalizednormal to dual basis mapping that can be determined prior to therun-time of the circuit 1306. Alternate embodiments could determine themapping after the circuit is powered up. In the present embodiment, thisnormal to dual basis mapping is hardwired in the circuit 1306. Inalternate embodiments, configurable logic such as programmable logicarrays or field programmable gate arrays could be used rather than hardwiring to implement the circuit 1306. The normal to dual basis mappingcan be determined using the trace function which can be defined as_(tr)(x)=x +_(x).spsb.2¹ +_(x).spsb.2² + . . . _(x).spsb.2^(m-1) whereTr(α^(j) λ_(i))=1 if i=j, and Tr(α^(j) λ_(i))=0 if i≠j. The word Z thatis provided to input 1304 of FIG. 13 is used to illustrate thederivation of the generalized normal to dual basis mapping for m=8.Generalized normal to dual basis mappings can be determined in a similarmanner for different values of m.

In particular, in the present embodiment where m=8, the word Z is thebyte Zbit₇ Zbit₆ Zbit₅ Zbit₄ Zbit₃ Zbit₂ Zbit₁ Zbit₀ where each of theterms Zbit_(x) =0 or 1 depending on the value of the word Z. Expressedusing the normal basis

    Z=Zbit.sub.7 α.sup.7 +Zbit.sub.6 α.sup.6 +Zbit.sub.5 α.sup.5 + . . . +Zbit.sub.0 α.sup.0.          (26)

It can be shown that the trace function also has the property that forany byte Z, Tr(Zα^(k))=Z_(d) bit_(k). Please see Hsu et al, The VLSIImplementation of a Reed-Solomon Encoder Using Berlekamp's Bit-SerialMultiplier Algorithm, IEEE Transactions on Computers, Vol. C-33, No. 10,October 1984. Using the equation Tr(Zα^(k))=Z_(d) bit_(k), a referencemapping can be determined between the normal basis and the dual basis.This reference mapping can be used to express each of the normal basiselements (e.g. α⁷, α⁶, . . . α⁰) in equation (26) in terms of dual basiselements (e.g.λ₇, λ₆, . . . λ₀). FIGS. 15A and 15B provide an example ofsuch a reference mapping 1500 for the present embodiment which uses theGalois field GF(2⁸).

FIGS. 15A and 15B illustrate in the reference mapping 1500 a power of αcolumn 1502, a normal basis column 1504 and a dual basis column 1506.The rows numbered 0 to 255 in the power of α column 1502 indicate thepower of α to which each row corresponds. For example, row 1508corresponds to α⁴⁹. The mapping between a particular power of α incolumn 1502 and its normal basis representation in column 1504 isdetermined using finite field arithmetic. Thus, the normal basisrepresentation for α⁴⁹ is determined using finite field arithmetic toexpress α⁴⁹ as a combination of powers of α where the powers are m-1 orless. The normal basis representation for α⁴⁹ is shown in column 1504 inthe same row as the 49 in column 1502.

The mapping between a particular power of α and its dual basisrepresentation in column 1506 is determined by substituting theparticular power of α in the equation Tr(Zα^(k))=Z_(d) bit_(k) as theelement Z. The dual basis representation of α⁴⁹ can be determined, forexample, by evaluating Tr(α⁴⁹ *α^(k))=Z_(d) bit_(k) where Z_(d) bit_(k)is the k^(th) bit of the dual basis equivalent of α⁴⁹. Varying k from 0to m-1 produces the 0 to m-1 position bits of the dual basis equivalentof α⁴⁹. This dual basis equivalent is shown in column 1506 in the rowhaving a 49 in the power of α column 1502. This dual basis equivalent isalso the dual basis equivalent of the normal basis representation of α⁴⁹shown in column 1504. Accordingly, the reference mapping 1500 enablesconversion between a power of α, its normal basis representation and itsdual basis representation. As shown in rows 1510, 1512, 1514, 1516,1518, 1520, 1522 and 1524 of FIGS. 15A and 15B, the dual basis consistsof a set of m linearly independent vectors having a one in only onedimension.

Thus, in the Galois field GF(2^(m)), each power of α having a power of mor more can be expressed in the normal basis representation as a sum ofthe normal basis elements (i.e. powers of α having powers from 0 tom-1). Expressing a power of α where the power of α is m or greater usingthe normal basis elements (powers of α from 0 to m-1) is illustrated inthe normal basis column 1504. In particular, each of the 0 bit to 7 bitcolumns in the normal basis column 1504 corresponds to a power of α inthe normal basis. For example, the 0 bit column correponds to α⁰, the 1bit column corresponds to α¹, . . . the 7 bit column corresponds to α⁷.The normal basis column illustrates which elements in the normal basisshould be summed to generate the power of α indicated in the power of αcolumn 1502. Thus, row 1508, which has a one in the 2 bit, 3 bit and 7bit columns, indicates that α⁴⁹ =α² +α³ +α⁷.

Similarly, the dual basis column 1506 illustrates which elements of thedual basis should be summed to generate the power of α indicated in thepower of α column 1502. In particular, column 1508 of row 49, having aone in the 4 bit and 6 bit columns of the dual basis column 1506indicates that α⁴⁹ =λ₄ +λ₆.

The reference mapping 1500 can also be used to determine which dualbasis elements should be summed to produce a normal basis element. Forexample, row 1510 corresponds to α⁰. From the dual basis column 1506 inthis row 1520 which has a one in the 5 bit column, it is clear that α⁰=λ₅. From the row 1512, it is clear that α¹ =λ₄. Similarly, it can bedetermined from the rows that have powers of α from 2 to 7 in the powerof α column 1502 that α² =λ₃ +λ₇, α³ =λ₂ +λ₆ +λ₇, α⁴ =λ₁ +λ₅ +λ₆ +λ₇, α⁵=λ₀ +λ₄ +λ₅ +λ₆, α⁶ =λ₃ +λ₄ +λ₅ and α⁷ =λ₂ +λ₃ +λ₄. These relations areused to produce the generalized normal to dual basis mapping.

Thus, substituting these values of the normal basis elements in terms ofthe dual basis elements into equation (26) produces the equation##EQU21## Factoring out each of the dual basis elements produces theequation ##EQU22## Comparing equation (28) to equation (25) (with m=8substituted in equation (25)), it can be determined that

Z_(d) bit₀ =Zbit₅ ;

Z_(d) bit₁ =Zbit₄ ;

Z_(d) bit₂ =Zbit₃ +Zbit₇

Z_(d) bit₃ =Zbit₂ +Zbit₆ +Zbit₇ ;

Z_(d) bit₄ =Zbit₁ +Zbit₅ +Zbit₆ +Zbit₇ ;

Z_(d) bit₅ =Zbit₀ +Zbit₄ +Zbit₅ +Zbit₆ ;

Z_(d) bit₆ =Zbit₃ +Zbit₄ +Zbit₅ ; and

Z_(d) bit₇ =Zbit₂ +Zbit₃ +Zbit₄

where each "+" sign represents mod(2) addition or an Exclusive OR. Inother words, the bits of the word Z_(d) can be determined from the bitsof the word Z. These equations together form the normal to dual basismapping. Each of the equations can be considered to be an individualmapping in the overall normal to dual basis mapping. In the presentembodiment, this normal to dual basis mapping is determined prior to runtime, although alternate embodiments need not do so. While the presentembodiment has determined mappings between a normal and dual basis forthe Galois field GF(2⁸), this technique can be generalized to devise anormal to dual basis converter for the Galois field GF(2^(m)) for any m.Such a normal to dual basis converter circuit can be implemented togenerate from any normal basis m bit word in the Galois field GF(2^(m))the corresponding dual basis word.

In the present embodiment, the normal to dual basis converter circuitimplements all of the individual mappings in parallel. For example, inFIG. 14A, the path 1406 implements the individual mapping Z_(d) bit₀=Zbit₅. The path 1408 implements the individual mapping Z_(d) bit₁=Zbit₄. The path 1410, which includes the two input Exclusive OR 1412,implements the individual mapping Z_(d) bit₂ =Zbit₃ +Zbit₇. FIG. 14Billustrates the path 1414, which includes the three input Exclusive OR1416, implementing the individual mapping Z_(d) bit₃ =Zbit₂ +Zbit₆+Zbit₇. FIG. 14C illustrates the path 1418, which includes the fourinput Exclusive OR 1420, implementing the individual mapping Z_(d) bit₄=Zbit₁ +Zbit₅ +Zbit₆ +Zbit₇. FIG. 14D illustrates the path 1422, whichincludes the four input Exclusive OR 1424, implementing the individualmapping Z_(d) bit₅ =Zbit₀ +Zbit₄ +Zbit₅ +Zbit₆. FIG. 14E illustrates thepath 1426, which includes the three input Exclusive OR 1428,implementing the individual mapping Z_(d) bit₆ =Zbit₃ +Zbit₄ +Zbit₅.FIG. 14F illustrates the path 1430, which includes the three inputExclusive OR 1432, implementing the individual mapping Z_(d) bit₇ =Zbit₂+Zbit₃ +Zbit₄. Implementing these paths and/or mappings in parallelresults in a normal to dual basis converter circuit that has a maximumdelay that corresponds to the delay of a four input Exclusive OR, suchas Exclusive OR 1424, for example. Alternate embodiments of theinvention could implement all of these paths 1406, 1408, 1410, 1414,1418, 1422, 1426 and 1430 in series or could implement some of the pathsin series and some of the paths in parallel. Implementing all of thepaths in parallel, however, typically will provide the greatest speed.Embodiments of the invention could also implement the mappings usingcircuits logically equivalent to those illustrated in FIGS. 14A-14F. Forexample, a five input Exclusive OR might be implemented as a pluralityof Exclusive OR circuits or some other logically equivalent circuit.

Referring back to FIG. 13, the multiplier circuit 1308 will now bedescribed in more detail. This circuit 1308 contains generator circuit1316 and vector multiplier circuit 1318. The generator circuit 1316receives the bits of the word Z_(d) and generates the "additional vectorbits" of the dual basis vectors Z_(d) bit_(m) . . . Z_(d) bit_(2m-2)(e.g. Z_(d) bit₈, Z_(d) bit₉ . . . Z_(d) bit₁₄ of FIGS. 16A and 16B).The bits of the words G and Z_(d) are supplied to the vector multiplier1318. The "additional vector" bits Z_(d) bit_(m) . . . Z_(d) bit_(2m-2)also are supplied from the output of generator circuit 1316 to thevector multiplier 1318. Vector multiplier 1318 executes themultiplications shown in FIG. 16C to generate the product GZ_(d). Inparticular, as shown in FIG. 16C, the binary mapping vector 1602 ismultiplied individually with each of the dual basis vectors 1604_(p) forp varying from 0 to m-1. Each individual multiplication produces therespective bits GZ_(d) bit_(p) of the dual basis product GZ_(d).

Similar to the normal to dual basis converter circuit 1306, thegenerator circuit 1316 uses exclusive OR gates to accomplish the mod(2)arithmetic that generates the elements Z_(d) bit₈, Z_(d) bit₉ . . .Z_(d) bit₁₄. As demonstrated in the discussion with reference to FIGS.16A and 16B, the following individual mappings between the bits of theword Z_(d) and the additional vector bits are true:

    Z.sub.d bit.sub.8 =Z.sub.d bit.sub.0 +Z.sub.d bit.sub.2 +Z.sub.d bit.sub.3 +Z.sub.d bit.sub.4,                                       (29)

    Z.sub.d bit.sub.9 =Z.sub.d bit.sub.1 +Z.sub.d bit.sub.3 +Z.sub.d bit.sub.4 +Z.sub.d bit.sub.5,                                       (30)

    Z.sub.d bit.sub.10 =Z.sub.d bit.sub.2 +Z.sub.d bit.sub.4 +Z.sub.d bit.sub.5 +Z.sub.d bit.sub.6,                                       (31)

    Z.sub.d bit.sub.11 =Z.sub.d bit.sub.3 +Z.sub.d bit.sub.5 +Z.sub.d bit.sub.6 +Z.sub.d bit.sub.7,                                       (32)

    Z.sub.d bit.sub.12 =Z.sub.d bit.sub.0 +Z.sub.d bit.sub.2 +Z.sub.d bit.sub.3 +Z.sub.d bit.sub.6 +Z.sub.d bit.sub.7,                    (33)

    Z.sub.d bit.sub.13 =Z.sub.d bit.sub.0 +Z.sub.d bit.sub.1 +Z.sub.d bit.sub.2 +Z.sub.d bit.sub.7,                                       (34)

    Z.sub.d bit.sub.14 =Z.sub.d bit.sub.0 +Z.sub.d bit.sub.1 +Z.sub.d bit.sub.4.(35)

In the present embodiment, equations (29)-(35) are implemented ingenerator circuit 1316 in the same manner as the individual mappings areimplemented in the normal to dual basis converter circuit 1306(discussed above) and in the dual to normal basis converter circuit 1312(discussed below). Again, each "+" sign is mod(2) arithmetic. Thus, eachof equations (29)-(32) and (34) is implemented using a four inputexclusive OR gate, equation (33) is implemented using a five inputexclusive OR gate, and equation (35) is implemented using a three inputexclusive OR gate. Again, equivalent logic may be used in alternateembodiments.

With reference again to FIG. 13, the vector multiplier 1318 uses ANDgates to implement the multiplications and uses exclusive OR gates toimplement the mod(2) additions required in the vector multiplicationshown in FIG. 16C. Thus, as shown in FIG. 16C, the binary mapping vector1602 is multiplied by the dual basis vector 1604₀. This multiplicationrequires that the equation

    GZ.sub.d bit.sub.0 =(Gbit.sub.0 *Z.sub.d bit.sub.0)+

    (Gbit.sub.1 *Z.sub.d bit.sub.1)+(Gbit.sub.2 *Z.sub.d bit.sub.2)

    +(Gbit.sub.3 *Z.sub.d bit.sub.3)+(Gbit.sub.4

    *Z.sub.d bit.sub.4)+(Gbit.sub.5 Z.sub.d bit.sub.5)+

    (Gbit.sub.6 *Z.sub.d bit.sub.6)+(Gbit.sub.7 *Z.sub.d bit.sub.7).

be implemented. This equation is implemented in vector multipliercircuit 1318 as shown in FIG. 18. Each of the products of this equation(shown in parentheses) is produced by providing the appropriatemultiplicand and multiplier bits to the inputs of one of the two inputAND gates 1802. The resulting eight products are mod(2) added togetherby providing the products to the inputs of the eight input exclusive ORgate 1804. The output of the 8 input exclusive OR gate provides the bitGZ_(d) bit₀ of the dual basis product GZ_(d). Again, alternate logicallyequivalent gate configurations could be used (e.g. the two input ANDgates could be replaced by a NAND gate and an inverter; the eight inputexclusive OR gate could be replaced by a plurality of two inputexclusive OR gates). The remaining bits GZ_(d) bit₁ to GZ_(d) bit₇ aregenerated by performing the remaining vector multiplications illustratedin FIG. 16C and implementing the vector multiplications in the samemanner using two input AND gates and eight input exclusive OR gates.Thus, the bit GZ_(d) bit₁ is generated by implementing the equation

    GZ.sub.d bit.sub.1 =(Gbit.sub.0 *Z.sub.d bit.sub.1)+(Gbit.sub.1 *Z.sub.d bit.sub.2)+(Gbit.sub.2 *Z.sub.d bit.sub.3)+(Gbit.sub.3 *Z.sub.d bit.sub.4)+(Gbit.sub.4 *Z.sub.d bit.sub.5)+(Gbit.sub.5 Z.sub.d bit.sub.6)+(Gbit.sub.6 *Z.sub.d bit.sub.7)+(Gbit.sub.7 *Z.sub.d bit.sub.8)

where Z_(d) bit₈ =Z_(d) bit₀ +Z_(d) bit₂ +Z_(d) bit₃ +Z_(d) bit₄. Thisequation may be implemented by using AND gates to form the products andan exclusive OR gate to perform the mod(2) additions as described withrespect to GZ_(d) bit₀ (not shown). The remaining equations to derivebits GZ_(d) bit₂ . . . GZ_(d) bit₇ are similar and are illustrated inFIG. 16C by the vector multiplications. These equations may beimplemented in the same manner (not shown). The resulting bits GZ_(d)bit₀, GZ_(d) bit₁ . . . GZ_(d) bit₇ of the dual basis product GZ_(d) areoutput from the vector multiplier circuit 1318 and provided to the ouputof the multiplier circuit 1308. Again, these individual mappings and/orcircuit paths may all be implemented in parallel, all be implemented inseries, or some implemented in parallel and some in series.

FIGS. 17A-G illustrate the dual to normal basis converter circuit 1312which receives the dual basis product GZ_(d) from the multiplier circuit1308. This circuit 1312 is implemented in a similar manner to the normalto dual basis converter circuit 1306. In particular, the dual to normalbasis converter circuit 1312 implements a generalized dual to normalbasis mapping that can be determined prior to run-time. In the presentembodiment, this dual to normal basis mapping is hardwired in thecircuit 1312. In alternate embodiments, configurable logic such asprogrammable logic arrays or field programmable gate arrays could beused rather than hard wiring to implement the circuit 1312. Again, themultiple FIGS. 17A-G are used for ease of illustration to illustrate themappings from the input pins to the output pins of the converter circuit1312. The bits of the word GZ_(d) are input to the inputs 1702. The bitsof the word GZ are output from the outputs 1704.

Similar to the normal to dual basis mapping, the dual to normal basismapping can be determined from the mapping 1500 of FIGS. 15A and 15B.The product GZ_(d) is used to illustrate the derivation of the dual tonormal basis mapping.

In particular, in the present embodiment where m=8, the product GZ_(d)is the byte GZ_(d) bit₇ GZ_(d) bit₆ GZ_(d) bit₅ GZ_(d) bit₄ GZ_(d) bit₃GZ_(d) bit₂ GZ_(d) bit₁ GZ_(d) bit₀ where each of the terms GZ_(d)bit_(x) =0 or 1 depending on the value of the word GZ_(d). Expressedusing the dual basis

    GZ.sub.d =GZ.sub.d bit.sub.7 λ.sub.7 +GZ.sub.d bit.sub.6 λ.sub.6 +GZ.sub.d bit.sub.5 λ.sub.5 + . . . +GZ.sub.d bit.sub.0 λ.sub.0.                                 (36)

Using mapping 1500 in a similar manner to that described earlier, thedual basis elements can be expressed in terms of the normal basiselements. In particular, from row 1514 in FIG. 15B, it is seen that λ₀=α⁰ +α² +α³ +α⁵ +α⁷. From row 1516 it can be seen that λ₁ =α³ +α⁴ +α⁶+α⁷. From row 1518 in FIG. 15A it can be seen that λ₂ =α⁰ +α⁶ +α⁷. Fromrow 1520 it can be seen that λ₃ =α⁰ +α¹ +α⁶. From rows 1512 and 1510,respectively, we have shown that λ₄ =α¹ and λ₅ =α⁰. From row 1524 inFIG. 15B it can be see that λ₆ =α¹ +α² +α³ +α⁷. From row 1522 it can beseen that λ₇ =α⁰ +α¹ +α² +α⁶. Substituting these equations into equation(36) produces the equation ##EQU23## Factoring out the normal basiselements produces the equation ##EQU24## Comparing equation (38) to thenormal basis representation of the product GZ (i.e. GZ=α⁰ (GZbit₀)+α¹(GZbit₁)+α² (GZbit₂)+α³ (GZbit₃)+α⁴ (GZbit₄)+α⁵ (GZbit₅)+α⁶ (GZbit₆)+α⁷(GZbit₇)) it can be seen that

    GZbit.sub.0 =GZ.sub.d bit.sub.0 +GZ.sub.d bit.sub.2 +GZ.sub.d bit.sub.3 +GZ.sub.d bit.sub.5 +GZ.sub.d bit.sub.7 ;                 (39)

    GZbit.sub.1 =GZ.sub.d bit.sub.3 +GZ.sub.d bit.sub.4 +GZ.sub.d bit.sub.6 +GZ.sub.d bit.sub.7 ;                                     (40)

    GZbit.sub.2 =GZ.sub.d bit.sub.0 +GZ.sub.d bit.sub.6 +GZ.sub.d bit.sub.7 ;(41)

    GZbit.sub.3 =GZ.sub.d bit.sub.0 +GZ.sub.d bit.sub.1 +GZ.sub.d bit.sub.6 ;(42)

    GZbit.sub.4 =GZ.sub.d bit.sub.1 ;                          (43)

    GZbit.sub.5 =GZ.sub.d bit.sub.0 ;                          (44)

    GZbit.sub.6 =GZ.sub.d bit.sub.1 +GZ.sub.d bit.sub.2 +GZ.sub.d bit.sub.3 +GZ.sub.d bit.sub.7 ;                                     (45)

    GZbit.sub.7 =GZ.sub.d bit.sub.0 +GZ.sub.d bit.sub.1 +GZ.sub.d bit.sub.2 +GZ.sub.d bit.sub.6                                       (46)

where each "+" sign represents mod(2) addition or an Exclusive OR. Inother words, the bits of the word GZ can be determined from the bits ofthe word GZ_(d). Equations (39) to (46) illustrate the dual to normalbasis mapping. Each of these equations is an individual mapping in theoverall dual to normal basis mapping. In the present embodiment, thedual to normal basis mapping is determined prior to run time althoughalternate embodiments need not do so. While the present embodiment hasdetermined mappings between a dual and normal basis for the Galois fieldGF(2⁸), this technique can be generalized to devise a dual to normalbasis converter for the Galois field GF(2^(m)) for any m. Such a dual tonormal basis converter circuit can be implemented to generate from anydual basis m bit word in the Galois field GF(2^(m)) the correspondingnormal basis word.

In the present embodiment, the dual to normal basis converter circuit1312 implements all of the individual mappings in parallel. For example,in FIG. 17A, the path 1706, which includes the five input Exclusive OR1708, implements the equation (39). In FIG. 17B, the path 1710, whichincludes the four input Exclusive OR 1712, implements the equation (40).In FIG. 17C, the path 1714, which includes the three input Exclusive OR1716, implements the equation (41). In FIG. 17D, the path 1718, whichincludes the three input Exclusive OR 1720, implements the equation(42). In FIG. 17E the paths 1722 and 1724 implement the equations (43)and (44), respectively. In FIG. 17F, the path 1726, which includes thefour input Exclusive OR 1728, implements the equation (45). In FIG. 17G,the path 1730, which includes the four input Exclusive OR 1732,implements the equation (46). Implementing these paths in parallelenables construction of a dual to normal basis converter circuit thathas a maximum delay that corresponds to the delay of a five inputExclusive OR, such as Exclusive OR 1708, for example. Alternateembodiments of the invention could implement all of these paths 1706,1710, 1714, 1718, 1722, 1724, 1726 and 1730 in series or could implementsome of the paths in series and some of the paths in parallel.Implementing all of the paths in parallel, however, typically willprovide the greatest speed. Embodiments of the invention could alsoimplement the mappings using circuits that are logically equivalent tothe illustrated circuits. For example, a five input Exclusive OR mightbe implemented as a plurality of Exclusive OR circuits or some otherlogically equivalent circuit.

The bits output on the outputs 1704 of FIGS. 17A-17G provide the word GZwhich is the normal basis product GZ. While the Galois field multiplierhas been disclosed in the context of a Reed-Solomon decoder, it is notlimited to use in the context of a Reed-Solomon decoder. For example, itcould be used in a Reed-Solomon encoder or in other applications thatinvolve Galois field multiplication.

While Applicant has described the invention in terms of specificembodiments, the invention is not limited to or by the disclosedembodiments. The Applicant's invention may be applied beyond theparticular systems mentioned as examples in this specification. Althougha variety of circuits have been described in this specifications,embodiments of the invention need not use all of the circuits describedherein. For example, even within the multiplier 1300, embodiments of theinvention may use only one of the normal to dual basis converter, themultiplier circuit, the generator circuit, the vector multiplier circuitor the dual to normal basis converter.

What is claimed is:
 1. λ-μ calculator for processing error correctionpolynomials in an error correction decoder designed to detect t errors,the λ-μ calculator comprising:a plurality of λ-μ circuits each having aλ input, a λ output, a μ input and a μ output wherein the plurality ofλ-μ circuits are coupled together to form a series of λ-μ circuits suchthat the λ output of each immediately preceding λ-μ circuit is coupledto the λ input of each immediately succeeding λ-μ circuit and the μoutput of each immediately preceding λ-μ circuit is coupled to the μinput of each immediately succeeding λ-μ circuit and wherein the λ inputof a first of the λ-μ circuits provides an initial λ input, the μ inputof the first λ-μ circuit provides an initial μ input, the λ output froma last of the λ-μ circuits provides a final λ output and the μ outputfrom the last λ-μ circuit provides a final μ output; in each λ-μ circuita switch having a λ switch input, a μ switch input, a λ' switch output,a μ' switch output and a switch control input wherein the switch isadapted to couple the λ switch input to the λ' switch output and the μswitch input to the μ' switch output when a first control signal ispresent at the switch control input, wherein the switch is adapted tocouple the λ switch input to the μ' switch output and the μ switch inputto the λ' switch output when a second control signal is present at theswitch control input and wherein the λ switch input is coupled to the λinput and the μ switch input is coupled to the μ input, in each λ-μcircuit a λ processing path coupled in series between the λ' switchoutput and a λ bypass multiplexor wherein the λ processing path isadapted to process error correction polynomials; in each λ-μ circuit a λprocessing bypass path coupled in series between the λ input and the λbypass multiplexor wherein the λ processing bypass path is adapted topass error correction polynomials unprocessed and wherein the λ bypassmultiplexor is adapted to couple alternatively the λ input to the λoutput through the λ processing bypass path and the λ' switch output tothe λ output through the λ processing path; in each λ-μ circuit a μprocessing path coupled in series between the μ' switch output and a μbypass multiplexor wherein the μ processing path is adapted to processerror correction polynomials; in each λ-μ circuit a μ processing bypasspath coupled in series between the μ input and the μ bypass multiplexorwherein the μ processing bypass path is adapted to pass error correctionpolynomials unprocessed and wherein the μ bypass multiplexor is adaptedto couple alternatively the μ input to the μ output through the μprocessing bypass path and the μ' switch output to the μ output throughthe μ processing path.
 2. The λ-μ calculator of claim 1, wherein the μprocessing path in each λ-μ circuit comprises:a μ' delay path, a μ'bypass path and a μ' multiplexor wherein the μ' delay path is coupled inseries with the μ' multiplexor between the μ' switch output and the μoutput and wherein the μ' bypass path is coupled in series with the μ'multiplexor between the μ' switch output and the μ output such that theμ' multiplexor is adapted to couple the μ' switch output to the μ outputthrough one of the μ' delay path and the μ' bypass path.
 3. The λ-μcalculator of claim 1, wherein the λ bypass multiplexor couples the λ'switch output to the λ output through the λ processing path when the μbypass multiplexor couples the μ' switch output to the μ output throughthe μ processing path.
 4. The λ-μ calculator of claim 1, furthercomprising:a delay element and a memory element coupled in seriesbetween the λ input and the λ switch input such that the delay elementand the memory element are also coupled in series with the λ processingbypass path between the λ input and the λ bypass multiplexor; and asecond delay element and a second memory element coupled in seriesbetween the μ input and the μ switch input such that the second delayelement and the second memory element are also coupled in series withthe μ processing bypass path between the μ input and the μ bypassmultiplexor.
 5. The λ-μ calculator of claim 1, whereinan R polynomial isassociated with a particular λ-μ circuit; a Q polynomial is associatedwith the particular λ-μ circuit; when a degree of one of the Rpolynomial and the Q polynomial that are associated with the particularλ-μ circuit is less than t, then the λ bypass multiplexor in theparticular λ-μ circuit couples the λ input to the λ output through the λprocessing bypass path and the μ bypass multiplexor in the particularλ-μ circuit couples the μ input to the μ output through the μ processingbypass path; when the degree of the R polynomial that is associated withthe particular λ-μ circuit is not less than t and when the degree of theQ polynomial that is associated with the particular λ-μ circuit is notless than t, then the λ bypass multiplexor in the particular λ-μ circuitcouples the λ input to the λ output through the λ processing path andthe μ bypass multiplexor in the particular λ-μ circuit couples the μinput to the μ output through the μ processing path.
 6. The λ-μcalculator of claim 1, wherein the λ bypass multiplexor couples the λinput to the λ output through the λ processing bypass path when the μbypass multiplexor couples the μ input to the μ output through the μprocessing bypass path.
 7. The λ-μ calculator of claim 6, wherein the λbypass multiplexor couples the λ' switch output to the λ output throughthe λ processing path when the μ bypass multiplexor couples the μ'switch output to the μ output through the μ processing path.
 8. The λ-μcalculator of claim 1, wherein the λ processing path in each λ-μ circuitcomprises:a λ' delay path, a λ' calculation path, a λ' multiplexor and adelay adjustment circuit wherein the λ' delay path is coupled in serieswith the λ' multiplexor, the delay adjustment circuit and the λ bypassmultiplexor between the λ' switch output and the λ output and whereinthe λ' calculation path is coupled in series with the λ' multiplexor,the delay adjustment circuit and the λ bypass multiplexor between the λ'switch output and the λ output such that the λ' multiplexor is adaptedto couple the λ' switch output to the λ output through one of the λ'delay path and the λ' calculation path.
 9. The λ-μ calculator of claim8, wherein a cross couple path is coupled between the μ processing pathand the λ' calculation path and wherein the λ' calculation pathcomprises:a multiplier having a first input, a second input and anoutput; a calculation path memory element having an output; an adderhaving a first input, a second input and an output wherein the λ' switchoutput is coupled to the first input of the multiplier, the output ofthe calculation path memory element is coupled to the second input ofthe multiplier, the output of the multiplier is coupled to the firstinput of the adder, the cross couple path is coupled to the second inputof the adder and the output of the adder is coupled to the λ'multiplexor.
 10. The λ-μ calculator of claim 9, wherein the cross couplepath comprises:a cross couple memory element having an output; amultiplier having a first input, a second input and an output whereinthe output of the cross couple memory element is coupled to the firstinput of the multiplier, the μ processing path is coupled to the secondinput of the multiplier and the output of the multiplier is coupled tothe second input of the adder.
 11. The λ-μ calculator of claim 10,whereinan R polynomial is associated with a particular λ-μ circuit; a Qpolynomial is associated with the particular λ-μ circuit; when a degreeof the R polynomial that is associated with the particular λ-μ circuitis no less than a degree of the Q polynomial that is associated with theparticular λ-μ circuit, then the switch in the particular λ-μ circuitcouples the λ switch input to the λ' switch output and the μ switchinput to the μ' switch output when the degree of the R polynomial thatis associated with the particular λ-μ circuit is less than the degree ofthe Q polynomial that is associated with the particular λ-μ circuit,then the switch in the particular λ-μ circuit couples the λ switch inputto the μ' switch output and the μ switch input to the λ' switch output;a polynomial output from the λ' switch output of the particular λ-μcircuit is the λ' polynomial for that particular λ-μ circuit and apolynomial output from the μ' switch output of the particular λ-μcircuit is the μ' polynomial for that particular λ-μ circuit; thecalculation path memory element in the particular λ-μ circuit stores theleading coefficient of the μ' polynomial and the cross couple memoryelement in the particular λ-μ circuit stores the leading coefficient ofthe λ' polynomial.
 12. The λ-μ calculator of claim 8, wherein the delayadjustment circuit in each λ-μ circuit comprises:a delay adjustmentmultiplexor; a delay element coupled to the delay adjustmentmultiplexor; a delay bypass path coupled to the delay adjustmentmultiplexor wherein the delay adjustment multiplexor is adapted tocouple the λ' switch output to the λ output through one of the delayelement and the delay bypass path.
 13. The λ-μ calculator of claim 12,wherein the delay element comprises:a memory element.
 14. The λ-μcalculator of claim 12, whereinan R polynomial is associated with aparticular λ-μ circuit; when a degree of the R polynomial associatedwith the particular λ-μ circuit is less than t, then the delayadjustment multiplexor couples the λ' switch output to the λ outputthrough the delay element; when the degree of the R polynomialassociated with the particular λ-μ circuit is not less than t, then thedelay adjustment multiplexor couples the λ' switch output to the λoutput through the delay bypass path.
 15. The λ-μ calculator of claim12, wherein the delay element provides a delay that is longer than adelay provided by the delay bypass path.
 16. The λ-μ calculator of claim15, wherein the delay element provides a delay that is at least onecycle longer than a delay provided by the delay bypass path.
 17. The λ-μcalculator of claim 8, wherein the μ processing path in each λ-μ circuitcomprises:a λ' delay path, a μ' bypass path and a μ' multiplexor whereinthe μ' delay path is coupled in series with the μ' multiplexor betweenthe μ' switch output and the μ output and wherein the μ' bypass path iscoupled in series with the μ' multiplexor between the μ' switch outputand the μ output such that the μ' multiplexor is adapted to couple theμ' switch output to the μ output through one of the μ' delay path andthe μ' bypass path; and in each λ-μ circuit a cross couple path coupledbetween the μ' delay path and the λ' calculation path.
 18. The λ-μcalculator of claim 17, wherein the cross couple path comprises:a crosscouple memory element having an output; a multiplier having a firstinput, a second input and an output wherein the output of the crosscouple memory element is coupled to the first input of the multiplier,the μ' delay path is coupled to the second input of the multiplier andthe output of the multiplier is coupled to the λ' calculation path. 19.The λ-μ calculator of claim 17, wherein the λ' multiplexor couples theλ' switch output to the λ output through the λ' delay path when the μ'multiplexor couples the μ' switch output to the μ output through the μ'bypass path.
 20. The λ-μ calculator of claims 17, wherein the λ' delaypath and the μ' delay path each comprise a memory element.
 21. The λ-μcalculator of claim 17, wherein the λ' bypass path, the μ' multiplexorand the λ bypass multiplexor are adapted to couple directly the μ'switch output to the μ output.
 22. The λ-μ calculator of claim 17,whereinan R polynomial is associated with a particular λ-μ circuit; a Qpolynomial is associated with the particular λ-μ circuit; when a degreeof the R polynomial that is associated with the particular λ-μ circuitis no less than a degree of the Q polynomial that is associated with theparticular λ-μ circuit, then the switch in the particular λ-μ circuitcouples the λ switch input to the λ' switch output and the μ switchinput to the μ' switch output; and when the degree of the R polynomialthat is associated with the particular λ-μ circuit is less than thedegree of the Q polynomial that is associated with the particular λ-μcircuit, then the switch in the particular λ-μ circuit couples the λswitch input to the μ' switch output and the μ switch input to the λ'switch output.
 23. The λ-μ calculator of claim 17, whereina first Qpolynomial and a second Q polynomial are associated with a particularλ-μ circuit; when a lead coefficient of the first Q polynomial that isassociated with the particular λ-μ circuit is zero and when a degree ofthe second Q polynomial that is associated with the particular λ-μcircuit is not less than t, then the λ' multiplexor couples the λ'switch output to the λ output through the λ' delay path and the μ'multiplexor couples the μ' switch output to the μ output through the μ'bypass path.
 24. The λ-μ calculator of claim 23, whereinwhen the leadcoefficient of the first Q polynomial that is associated with theparticular λ-μ circuit is not zero or when the degree of the second Qpolynomial that is associated with the particular λ-μ circuit is lessthan t, the λ' multiplexor couples the λ' switch output to the λ outputthrough the λ' calculation path and the μ' multiplexor couples the μ'switch output to the μ output through the μ' delay path.
 25. The λ-μcalculator of claim 17, wherein the λ' multiplexor couples the λ' switchoutput to the λ output through the λ' calculation path when the μ'multiplexor couples the μ' switch output to the μ output through the μ'delay path.
 26. The λ-μ calculator of claim 25, wherein the μ' delaypath introduces a delay that is greater than a delay introduced by theλ' calculation path.
 27. The λ-μ calculator of claim 26, wherein thedelay introduced by the μ' delay path is at least one cycle longer thanthe delay introduced by the λ' calculation path.
 28. The λ-μ calculatorof claim 27, wherein the μ' delay path comprises a memory element. 29.The λ-μ calculator of claim 25, wherein the λ' multiplexor couples theλ' switch output to the λ output through the λ' delay path when the μ'multiplexor couples the μ' switch output to the μ output through the μ'bypass path.
 30. The λ-μ calculator of claim 29, wherein the λ' delaypath introduces a delay that is longer than a delay introduced by the μ'bypass path.
 31. The λ-μ calculator of claim 30, wherein the delayintroduced by the λ' delay path is at least one cycle longer than thedelay introduced by the μ' bypass path.
 32. The λ-μ calculator of claim31, wherein the λ' delay path comprises a memory element.